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divisible by 2. Conjecture: If a number is divisible by 4, then it is even. Let x, y, and z represent the following. x : A number is divisible by 4. y : A number is divisible by 2. z : A number is even. You are given that x → y and z → y. The Law of Syllogism cannot be used to draw a conclusion since y is the conclusion of both conditionals. Even though the conjecture x → z is true, the logic used to draw the conclusion is not valid. 3. Determine if the conjecture is valid by the Law of Syllogism. Given: If an animal is a mammal, then it has hair. If an animal is a dog, then it is a mammal. Conjecture: If an animal is a dog, then it has hair. E X A M P L E 4 Applying the Laws of Deductive Reasoning Draw a conclusion from the given information. A Given: If a team wins 10 games, then they play in the finals. If a team plays in the finals, then they travel to Boston. The Ravens won 10 games. Conclusion: The Ravens will travel to Boston. B Given: If two angles form a linear pair, then they are adjacent. If two angles are adjacent, then they share a side. ∠1 and ∠2 form a linear pair. Conclusion: ∠1 and ∠2 share a side. 4. Draw a conclusion from the given information. Given: If a polygon is a triangle, then it has three sides. If a polygon has three sides, then it is not a quadrilateral. Polygon P is a triangle. THINK AND DISCUSS 1. Could “A square has exactly two sides” be the conclusion of a valid argument? If so, what do you know about the truth value of the given information? 2. Explain why writing conditional statements as symbols might help you evaluate the validity of an argument. 3. GET ORGANIZED Copy and complete the graphic organizer. Write each law in your own words and give an example of each. 90 90 Chapter 2 Geometric Reasoning �� 2-3 Exercises Exercises KEYWORD: MG7 2-3 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Explain how deductive reasoning differs from inductive reasoning Does each conclusion
use inductive or deductive reasoning? p. 88 2. At Bell High School, students must take Biology before they take Chemistry. Sam is in Chemistry, so Marcia concludes that he has taken Biology. 3. A detective learns that his main suspect was out of town the day of the crime. He concludes that the suspect is innocent Determine if each conjecture is valid by the Law of Detachment. p. 89 4. Given: If you want to go on a field trip, you must have a signed permission slip. Zola has a signed permission slip. Conjecture: Zola wants to go on a field trip. 5. Given: If the side lengths of a rectangle are 3 ft and 4 ft, then its area is 12 ft 2. A rectangle has side lengths of 3 ft and 4 ft. Conjecture: The area of the rectangle is 12 ft Determine if each conjecture is valid by the Law of Syllogism. p. 89 6. Given: If you fly from Texas to California, you travel from the central to the Pacific time zone. If you travel from the central to the Pacific time zone, then you gain two hours. Conjecture: If you fly from Texas to California, you gain two hours. 7. Given: If a figure is a square, then the figure is a rectangle. If a figure is a square, then it is a parallelogram. Conjecture: If a figure is a parallelogram, then it is a rectangle. 90 8. Draw a conclusion from the given information. Given: If you leave your car lights on overnight, then your car battery will drain. If your battery is drained, your car might not start. Alex left his car lights on last night. Independent Practice Does each conclusion use inductive or deductive reasoning? PRACTICE AND PROBLEM SOLVING For See Exercises Example 9–10 11 12 13 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S6 Application Practice p. S29 9. The sum of the angle measures of a triangle is 180°. Two angles of a triangle measure 40° and 60°, so Kandy concludes that the third angle measures 80°. 10. All of the students in Henry’s Geometry class are juniors. Alexander takes Geometry, but has another teacher. Henry concludes that Alexander is also a junior. 11. Determine if the conjecture is valid by the Law of Detachment
. Given: If one integer is odd and another integer is even, their product is even. The product of two integers is 24. Conjecture: One of the two integers is odd. 2- 3 Using Deductive Reasoning to Verify Conjectures 91 91 � 12. Science Determine if the conjecture is valid by the Law of Syllogism. Given: If an element is an alkali metal, then it reacts with water. If an element is in the first column of the periodic table, then it is an alkali metal. Conjecture: If an element is in the first column of the periodic table, then it reacts with water. 13. Draw a conclusion from the given information. Given: If Dakota watches the news, she is informed about current events. If Dakota knows about current events, she gets better grades in Social Studies. Dakota watches the news. 14. Technology Joseph downloads a file in 18 minutes with a dial-up modem. How long would it take to download the file with a Cheetah-Net cable modem? Recreation Recreation Use the true statements below for Exercises 15–18. Determine whether each conclusion is valid. The Top Thrill Dragster is 420 feet tall and includes a 400-foot vertical drop. It twists 270° as it drops. It is one of 16 roller coasters at Cedar Point amusement park. I. The Top Thrill Dragster is at Cedar Point amusement park in Sandusky, OH. II. Carter and Mary go to Cedar Point. III. The Top Thrill Dragster roller coaster reaches speeds of 120 mi/h. IV. When Carter goes to an amusement park, he rides all the roller coasters. 15. Carter went to Sandusky, OH. 16. Mary rode the Top Thrill Dragster. 17. Carter rode a roller coaster that travels 120 mi/h. 18. Mary rode a roller coaster that travels 120 mi/h. 19. Critical Thinking Is the argument below a valid application of the Law of Syllogism? Is the conclusion true? Explain your answers. If 3 - x < 5, then x < -2. If x < -2, then -5x > 10. Thus, if 3 - x < 5, then -5x > 10. 20. /////ERROR ANALYSIS///// Below are two conclusions. Which is incorrect? Explain the error. If two angles are complementary, their measures add to 90°. If an angle measures 90°,
then it is a right angle. ∠A and ∠B are complementary. 21. Write About It Write one example of a real-life logical argument that uses the Law of Detachment and one that uses the Law of Syllogism. Explain why the conclusions are valid. 22. This problem will prepare you for the Multi-Step TAKS Prep on page 102. When Alice meets the Pigeon in Wonderland, the Pigeon thinks she is a serpent. The Pigeon reasons that serpents eat eggs, and Alice confirms that she has eaten eggs. a. Write “Serpents eat eggs” as a conditional statement. b. Is the Pigeon’s conclusion that Alice is a serpent valid? Explain your reasoning. 92 92 Chapter 2 Geometric Reasoning �� 23. The Supershots scored over 75 points in each of ten straight games. The newspaper predicts that they will score more than 75 points tonight. Which form of reasoning is this conclusion based on? Deductive reasoning, because the conclusion is based on logic Deductive reasoning, because the conclusion is based on a pattern Inductive reasoning, because the conclusion is based on logic Inductive reasoning, because the conclusion is based on a pattern 24.  HF bisects ∠EHG. Which conclusion is NOT valid? E, F, and G are coplanar. ∠EHF ≅ ∠FHG ̶̶ EF ≅ ̶̶ FG m∠EHF = m∠FHG 25. Gridded Response If Whitney plays a low G on her piano, the frequency of the note is 24.50 hertz. The frequency of a note doubles with each octave. What is the frequency in hertz of a G note that is 3 octaves above low G? CHALLENGE AND EXTEND 26. Political Science To be eligible to hold the office of the president of the United States, a person must be at least 35 years old, be a natural-born U.S. citizen, and have been a U.S. resident for at least 14 years. Given this information, what conclusion, if any, can be drawn from the statements below? Explain your reasoning. Andre is not eligible to be the president of the United States. Andre has lived in the United States for 16 years. 27
. Multi-Step Consider the two conditional statements below. If you live in San Diego, then you live in California. If you live in California, then you live in the United States. a. Draw a conclusion from the given conditional statements. b. Write the contrapositive of each conditional statement. c. Draw a conclusion from the two contrapositives. d. How does the conclusion in part a relate to the conclusion in part c? 28. If Cassie goes to the skate park, Hanna and Amy will go. If Hanna or Amy goes to the skate park, then Marc will go. If Marc goes to the skate park, then Dallas will go. If only two of the five people went to the skate park, who were they? SPIRAL REVIEW Simplify each expression. (Previous course) 29. 2 (x + 5) 30. (4y + 6) - (3y - 5) 31. (3c + 4c) + 2 (-7c + 7) Find the coordinates of the midpoint of the segment connecting each pair of points. (Lesson 1-6) 32. (1, 2) and (4, 5) 33. (-3, 6) and (0, 1) 34. (-2.5, 9) and (2.5, -3) Identify the hypothesis and conclusion of each conditional statement. (Lesson 2-2) 35. If the fire alarm rings, then everyone should exit the building. 36. If two different lines intersect, then they intersect at exactly one point. 37. The statement ̶̶ AB ≅ ̶̶ CD implies that AB = CD. 2- 3 Using Deductive Reasoning to Verify Conjectures 93 93 �� 2-3 Solve Logic Puzzles In Lesson 2-3, you used deductive reasoning to analyze the truth values of conditional statements. Now you will learn some methods for diagramming conditional statements to help you solve logic puzzles. Use with Lesson 2-3 TEKS G.4.A Geometric structure: select an appropriate representation... in order to solve problems Activity 1 Bonnie, Cally, Daphne, and Fiona own a bird, cat, dog, and fish. No girl has a type of pet that begins with the same letter as her name. Bonnie is allergic to animal fur. Daphne feeds Fiona’s bird when Fiona is away. Make a table to determine who owns which animal. 1 Since no girl has a type of
pet that starts with the same letter as her name, place an X in each box along the diagonal of the table. 2 Bonnie cannot have a cat or dog because of her allergy. So she must own the fish, and no other girl can have the fish. Bird Cat Dog Fish Bird Cat Dog Fish Bonnie × Cally Daphne Fiona × × × Bonnie × Cally Daphne Fiona × × × × ✓ × × × 3 Fiona owns the bird, so place a check in 4 Therefore, Daphne owns the cat, and Cally Fiona’s row, in the bird column. Place an X in the remaining boxes in the same column and row. owns the dog. Bird Cat Dog Fish Bird Cat Dog Fish Bonnie Cally Daphne Fiona × × × ✓ × × × × × × ✓ × × × Bonnie Cally Daphne Fiona × × × ✓ × × ✓ × × ✓ × × ✓ × × × Try This 1. After figuring out that Fiona owns the bird in Step 3, why can you place an X in every other box in that row and column? 2. Ally, Emily, Misha, and Tracy go to a dance with Danny, Frank, Jude, and Kian. Ally and Frank are siblings. Jude and Kian are roommates. Misha does not know Kian. Emily goes with Kian’s roommate. Tracy goes with Ally’s brother. Who went to the dance with whom? Ally Emily Misha Tracy 94 94 Chapter 2 Geometric Reasoning Danny Frank Jude Kian Activity 2 A farmer has a goat, a wolf, and a cabbage. He wants to transport all three from one side of a river to the other. He has a boat, but it has only enough room for the farmer and one thing. The wolf will eat the goat if they are left alone together, and the goat will eat the cabbage if they are left alone. How can the farmer get everything to the other side of the river? You can use a network to solve this kind of puzzle. A network is a diagram of vertices and edges, also known as a graph. An edge is a curve or a segment that joins two vertices of the graph. A vertex is a point on the graph. 1 Let F represent the farmer, W represent the wolf, G represent the goat, and C represent the cabbage. Use an ordered pair to represent what is on each side of the river. The first ordered pair is (FWGC, —),
and the desired result is (—, FWGC). 2 Draw a vertex and label it with the first ordered pair. Then draw an edge and vertex for each possible trip the farmer could make across the river. If at any point a path results in an unworkable combination of things, no more edges can be drawn from that vertex. 3 From each workable vertex, continue to draw edges and vertices that represent the next trip across the river. When you get to a vertex for (—, FWGC), the network is complete. 4 Use the network to write out the solution in words. Try This 3. What combinations are unworkable? Why? 4. How many solutions are there to the farmer’s transport problem? How many steps does each solution take? 5. What is the advantage of drawing a complete solution network rather than working out one solution with a diagram? 6. Madeline has two measuring cups—a 1-cup measuring cup and a 3__ 4 -cup measuring cup. Neither cup has any markings on it. How can Madeline get exactly 1 __ 2 cup of flour in the larger measuring cup? Complete the network below to solve the problem. 2- 3 Geometry Lab 95 95 �� 2-4 Biconditional Statements and Definitions TEKS G.3.A Geometric structure: determine the validity of a conditional statement, its converse, inverse, and contrapositive. Also G.3.B Objective Write and analyze biconditional statements. Vocabulary biconditional statement definition polygon triangle quadrilateral Who uses this? A gardener can plan the color of the hydrangeas she plants by checking the pH of the soil. The pH of a solution is a measure of the concentration of hydronium ions in the solution. If a solution has a pH less than 7, it is an acid. Also, if a solution is an acid, it has a pH less than 7. When you combine a conditional statement and its converse, you create a biconditional statement. A biconditional statement is a statement that can be written in the form “p if and only if q.” This means “if p, then q” and “if q, then p.” The biconditional “p
if and only if q” can also be written as “p iff q” or p ↔ q. So you can define an acid with the following biconditional statement: A solution is an acid if and only if it has a pH less than 7. E X A M P L E 1 Identifying the Conditionals within a Biconditional Statement Write the conditional statement and converse within each biconditional. A Two angles are congruent if and only if their measures are equal. Let p and q represent the following. p : Two angles are congruent. q : Two angle measures are equal. The two parts of the biconditional p ↔ q are p → q and q → p. Conditional: If two angles are congruent, then their measures are equal. Converse: If two angle measures are equal, then the angles are congruent. B A solution is a base ↔ it has a pH greater than 7. Let x and y represent the following. x : A solution is a base. y : A solution has a pH greater than 7. The two parts of the biconditional x ↔ y are x → y and y → x. Conditional: If a solution is a base, then it has a pH greater than 7. Converse: If a solution has a pH greater than 7, then it is a base. Write the conditional statement and converse within each biconditional. 1a. An angle is acute iff its measure is greater than 0° and less than 90°. 1b. Cho is a member if and only if he has paid the $5 dues. 96 96 Chapter 2 Geometric Reasoning ��meansandProject TitleGeometry 2007 Student EditionSpec Numberge07sec01l04002aCreated ByKrosscore CorporationCreation Date10/07/2004 E X A M P L E 2 Writing a Biconditional Statement For each conditional, write the converse and a biconditional statement. A If 2x + 5 = 11, then x = 3. Converse: If x = 3, then 2x + 5 = 11. Biconditional: 2x + 5 = 11 if and only if x = 3. B If a point is a midpoint, then it divides the segment into two congruent segments. Converse: If a point divides a segment into two congruent segments
, then the point is a midpoint. Biconditional: A point is a midpoint if and only if it divides the segment into two congruent segments. For each conditional, write the converse and a biconditional statement. 2a. If the date is July 4th, then it is Independence Day. 2b. If points lie on the same line, then they are collinear. For a biconditional statement to be true, both the conditional statement and its converse must be true. If either the conditional or the converse is false, then the biconditional statement is false. E X A M P L E 3 Analyzing the Truth Value of a Biconditional Statement Determine if each biconditional is true. If false, give a counterexample. A A square has a side length of 5 if and only if it has an area of 25. Conditional: If a square has a side length of 5, then it has an area of 25. The conditional is true. Converse: If a square has an area of 25, then it has a side length of 5. The converse is true. Since the conditional and its converse are true, the biconditional is true. B The number n is a positive integer ↔ 2n is a natural number. Conditional: If n is a positive integer, then 2n is a natural number. The conditional is true. Converse: If 2n is a natural number, then n is a positive integer. The converse is false. If 2n = 1, then n = 1 __ 2, which is not an integer. Because the converse is false, the biconditional is false. Determine if each biconditional is true. If false, give a counterexample. 3a. An angle is a right angle iff its measure is 90°. 3b. y = -5 ↔ y 2 = 25 In geometry, biconditional statements are used to write definitions. A definition is a statement that describes a mathematical object and can be written as a true biconditional. Most definitions in the glossary are not written as biconditional statements, but they can be. The “if and only if” is implied. 2- 4 Biconditional Statements and Definitions 97 97 In the glossary, a polygon is defined as a closed plane figure formed by three or
more line segments. Each segment intersects exactly two other segments only at their endpoints, and no two segments with a common endpoint are collinear. Polygons Not Polygons A triangle is defined as a three-sided polygon, and a quadrilateral is a four-sided polygon. A good, precise definition can be used forward and backward. For example, if a figure is a quadrilateral, then it is a four-sided polygon. If a figure is a four-sided polygon, then it is a quadrilateral. To make sure a definition is precise, it helps to write it as a biconditional statement. E X A M P L E 4 Writing Definitions as Biconditional Statements Write each definition as a biconditional. A A triangle is a three-sided polygon. Think of definitions as being reversible. Postulates, however, are not necessarily true when reversed. A figure is a triangle if and only if it is a three-sided polygon. B A segment bisector is a ray, segment, or line that divides a segment into two congruent segments. A ray, segment, or line is a segment bisector if and only if it divides a segment into two congruent segments. Write each definition as a biconditional. 4a. A quadrilateral is a four-sided polygon. 4b. The measure of a straight angle is 180°. THINK AND DISCUSS 1. How do you determine if a biconditional statement is true or false? 2. Compare a triangle and a quadrilateral. 3. GET ORGANIZED Copy and complete the graphic organizer. Use the definition of a polygon to write a conditional, converse, and biconditional in the appropriate boxes. 98 98 Chapter 2 Geometric Reasoning �� 2-4 Exercises Exercises KEYWORD: MG7 2-4 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary How is a biconditional statement different from a conditional statement Write the conditional statement and converse within each biconditional. p. 96 2. Perry can paint the entire living room if and only if he has enough paint. 3. Your medicine will be ready by 5 P.M. if and only if you drop your prescription off by 8 A.M
For each conditional, write the converse and a biconditional statement. p. 97 4. If a student is a sophomore, then the student is in the tenth grade. 5. If two segments have the same length, then they are congruent. 97 Multi-Step Determine if each biconditional is true. If false, give a counterexample. 6. xy = 0 ↔ x = 0 or y = 0. 7. A figure is a quadrilateral if and only if it is a polygon Write each definition as a biconditional. p. 98 8. Parallel lines are two coplanar lines that never intersect. 9. A hummingbird is a tiny, brightly colored bird with narrow wings, a slender bill, and a long tongue. Independent Practice Write the conditional statement and converse within each biconditional. PRACTICE AND PROBLEM SOLVING For See Exercises Example 10–12 13–15 16–17 18–19 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S6 Application Practice p. S29 10. Three points are coplanar if and only if they lie in the same plane. 11. A parallelogram is a rectangle if and only if it has four right angles. 12. A lunar eclipse occurs if and only if Earth is between the Sun and the Moon. For each conditional, write the converse and a biconditional statement. 13. If today is Saturday or Sunday, then it is the weekend. 14. If Greg has the fastest time, then he wins the race. 15. If a triangle contains a right angle, then it is a right triangle. Multi-Step Determine if each biconditional is true. If false, give a counterexample. 16. Felipe is a swimmer if and only if he is an athlete. 17. The number 2n is even if and only if n is an integer. Write each definition as a biconditional. 18. A circle is the set of all points in a plane that are a fixed distance from a given point. 19. A catcher is a baseball player who is positioned behind home plate and who catches throws from the pitcher. 2- 4 Biconditional Statements and Definitions 99 99 Algebra Determine if a true biconditional can be written from each conditional statement. If not, give a counterexample. 20
. If a = b, then ⎜a⎟ = ⎜b⎟. x + 8 = 12. 21. If 3x - 2 = 13, then 4 _ 5 23. If x > 0, then x 2 > 0. 22. If y 2 = 64, then 3y = 24. Use the diagrams to write a definition for each figure. 24. 25. Biology 26. Biology White blood cells are cells that defend the body against invading organisms by engulfing them or by releasing chemicals called antibodies. Write the definition of a white blood cell as a biconditional statement. White blood cells live less than a few weeks. A drop of blood can contain anywhere from 7000 to 25,000 white blood cells. Explain why the given statement is not a definition. 27. An automobile is a vehicle that moves along the ground. 28. A calculator is a machine that performs computations with numbers. 29. An angle is a geometric object formed by two rays. Chemistry Use the table for Exercises 30–32. Determine if a true biconditional statement can be written from each conditional. 30. If a solution has a pH of 4, then it is tomato juice. 31. If a solution is bleach, then its pH is 13. 32. If a solution has a pH greater than 7, then it is not battery acid. pH 0 4 6 8 13 14 Examples Battery Acid Acid rain, tomato juice Saliva Sea water Bleach, oven cleaner Drain cleaner Complete each statement to form a true biconditional. 33. The circumference of a circle is 10π if and only if its radius is?. ̶̶̶ 34. Four points in a plane form a? if and only if no three of them are collinear. ̶̶̶ 35. Critical Thinking Write the definition of a biconditional statement as a biconditional statement. Use the conditional and converse within the statement to explain why your biconditional is true. 36. Write About It Use the definition of an angle bisector to explain what is meant by the statement “A good definition is reversible.” 37. This problem will prepare you for the Multi-Step TAKS Prep on page 102. a. Write “I say what I mean” and “I mean what I say” as conditionals. b. Explain why the biconditional statement implied by Alice is false. “Then you should
say what you mean,” the March Hare went on. “I do,” Alice hastily replied; “at least—at least I mean what I say—that’s the same thing, you know.” 100 100 Chapter 2 Geometric Reasoning �� 38. Which is a counterexample for the biconditional “An angle measures 80° if and only if the angle is acute”? m∠S = 60° m∠S = 115° m∠S = 90° m∠S = 360° 39. Which biconditional is equivalent to the spelling phrase “I before E except after C”? The letter I comes before E if and only if I follows C. The letter E comes before I if and only if E follows C. The letter E comes before I if and only if E comes before C. The letter I comes before E if and only if I comes before C. 40. Which conditional statement can be used to write a true biconditional? If a number is divisible by 4, then it is even. If a ratio compares two quantities measured in different units, the ratio is a rate. If two angles are supplementary, then they are adjacent. If an angle is right, then it is not acute. 41. Short Response Write the two conditional statements that make up the biconditional “You will get a traffic ticket if and only if you are speeding.” Is the biconditional true or false? Explain your answer. CHALLENGE AND EXTEND 42. Critical Thinking Describe what the Venn diagram of a true biconditional statement looks like. How does this support the idea that a definition can be written as a true biconditional? 43. Consider the conditional “If an angle measures 105°, then the angle is obtuse.” a. Write the inverse of the conditional statement. b. Write the converse of the inverse. c. How is the converse of the inverse related to the original conditional? d. What is the truth value of the biconditional statement formed by the inverse of the original conditional and the converse of the inverse? Explain. 44. Suppose A, B, C, and D are coplanar, and A, B, and C are not collinear. What is the truth
value of the biconditional formed from the true conditional “If m∠ABD + m∠DBC = m∠ABC, then D is in the interior of ∠ABC”? Explain. 45. Find a counterexample for “n is divisible by 4 if and only if n 2 is even.” SPIRAL REVIEW Describe how the graph of each function differs from the graph of the parent function y = x 2. (Previous course) 46 47. y = -2 x 2 - 1 48. y = (x - 2) (x + 2) A transformation maps S onto T and X onto Y. Name each of the following. (Lesson 1-7) 49. the image of S 50. the image of X 51. the preimage of T Determine if each conjecture is true. If not, give a counterexample. (Lesson 2-1) 52. If n ≥ 0, then n _ 2 54. The vertices of the image of a figure under the translation (x, y) → (x + 0, y + 0) 53. If x is prime, then x + 2 is also prime. > 0. have the same coordinates as the preimage. 2- 4 Biconditional Statements and Definitions 101 101 SECTION 2A Inductive and Deductive Reasoning Rhyme or Reason Alice’s Adventures in Wonderland originated as a story told by Charles Lutwidge Dodgson (Lewis Carroll) to three young traveling companions. The story is famous for its wordplay and logical absurdities. 1. When Alice first meets the Cheshire Cat, she asks what sort of people live in Wonderland. The Cat explains that everyone in Wonderland is mad. What conjecture might the Cat make since Alice, too, is in Wonderland? 2. “I don’t much care where—” said Alice. “Then it doesn’t matter which way you go,” said the Cat. “—so long as I get somewhere,” Alice added as an explanation. “Oh, you’re sure to do that,” said the Cat, “if you only walk long enough.” Write the conditional statement implied by the Cat’s response to Alice. 3. “Well, then,” the Cat went on, “you see a dog growls when it’s
angry, and wags its tail when it’s pleased. Now I growl when I’m pleased, and wag my tail when I’m angry. Therefore I’m mad.” Is the Cat’s conclusion valid by the Law of Detachment or the Law of Syllogism? Explain your reasoning. 4. “You might just as well say,” added the Dormouse, who seemed to be talking in his sleep, “that ‘I breathe when I sleep’ is the same thing as ‘I sleep when I breathe’!” Write a biconditional statement from the Dormouse’s example. Explain why the biconditional statement is false. 102 102 Chapter 2 Geometric Reasoning Quiz for Lessons 2-1 Through 2-4 2-1 Using Inductive Reasoning to Make Conjectures Find the next item in each pattern. 1. 1, 10, 18, 25. July, May, March, … 3. 1 _ 4 2 8 5. A biologist recorded the following data about the weight of male lions in a wildlife park in Africa. Use the table to make a conjecture about the average weight of a male lion.,... 6. Complete the conjecture “The sum of two negative numbers is?.” ̶̶̶̶ 7. Show that the conjecture “If an even number is divided by 2, then the result is an even number” is false by finding a counterexample. SECTION 2A 4. ∣,,,... ID Number Weight (lb) A1902SM A1904SM A1920SM A1956SM A1974SM 387.2 420.5 440.6 398.7 415.0 2-2 Conditional Statements 8. Identify the hypothesis and conclusion of the conditional statement “An angle is obtuse if its measure is 107°.” Write a conditional statement from each of the following. 9. A whole number is 10. an integer. 11. The diagonals of a square are congruent. Determine if each conditional is true. If false, give a counterexample. 12. If an angle is acute, then it has a measure of 30°. 13. If 9x - 11 = 2x + 3, then x = 2. 14. Write the converse, inverse, and contrapositive of the statement “If
a number is even, then it is divisible by 4.” Find the truth value of each. 2-3 Using Deductive Reasoning to Verify Conjectures 15. Determine if the following conjecture is valid by the Law of Detachment. Given: If Sue finishes her science project, she can go to the movie. Sue goes to the movie. Conjecture: Sue finished her science project. 16. Use the Law of Syllogism to draw a conclusion from the given information. Given: If one angle of a triangle is 90°, then the triangle is a right triangle. If a triangle is a right triangle, then its acute angle measures are complementary. 2-4 Biconditional Statements and Definitions 17. For the conditional “If two angles are supplementary, the sum of their measures is 180°,” write the converse and a biconditional statement. 18. Determine if the biconditional “ √  x = 4 if and only if x = 16” is true. If false, give a counterexample. Ready to Go On? 103 103 �� 2-5 Algebraic Proof TEKS G.3.E Geometric structure: use deductive reasoning to prove a statement. Also G.3.B, G.3.C Objectives Review properties of equality and use them to write algebraic proofs. Identify properties of equality and congruence. Vocabulary proof The Distributive Property states that a (b + c) = ab + ac. Who uses this? Game designers and animators solve equations to simulate motion. (See Example 2.) A proof is an argument that uses logic, definitions, properties, and previously proven statements to show that a conclusion is true. If you’ve ever solved an equation in Algebra, then you’ve already done a proof! An algebraic proof uses algebraic properties such as the properties of equality and the Distributive Property. Properties of Equality Properties of Equality Addition Property of Equality If a = b, then a + c = b + c. Subtraction Property of Equality If a = b, then a - c = b - c. Multiplication Property of Equality Division Property of Equality Reflexive Property of Equality If a = b, then ac = bc. If a = b and c ≠ 0, then a __ c = b __ c. a = a Symmetric Property of Equality If a =
b, then b = a. Transitive Property of Equality If a = b and b = c, then a = c. Substitution Property of Equality If a = b, then b can be substituted for a in any expression. As you learned in Lesson 2-3, if you start with a true statement and each logical step is valid, then your conclusion is valid. An important part of writing a proof is giving justifications to show that every step is valid. For each justification, you can use a definition, postulate, property, or a piece of information that is given. E X A M P L E 1 Solving an Equation in Algebra Solve the equation -5 = 3n + 1. Write a justification for each step. -5 = 3n + 1 - 1 - 1 ̶̶̶̶̶ ̶̶̶ -6 = 3n _ _ -6 3n 3 3 -2 = n n = -2 = Given equation Subtraction Property of Equality Simplify. Division Property of Equality Simplify. Symmetric Property of Equality 1. Solve the equation 1 _ 2 t = -7. Write a justification for each step. 104 104 Chapter 2 Geometric Reasoning E X A M P L E 2 Problem-Solving Application To simulate the motion of an object in a computer game, the designer uses the formula sr = 3.6p to find the number of pixels the object must travel during each second of animation. In the formula, s is the desired speed of the object in kilometers per hour, r is the scale of pixels per meter, and p is the number of pixels traveled per second. The graphics in a game are based on a scale of 6 pixels per meter. The designer wants to simulate a vehicle moving at 75 km/h. How many pixels must the vehicle travel each second? Solve the equation for p and justify each step. Understand the Problem The answer will be the number of pixels traveled per second. List the important information: • sr = 3.6p • p: pixels traveled per second • s = 75 km/h • r = 6 pixels per meter Make a Plan Substitute the given information into the formula and solve. Solve sr = 3.6p (75) (6) = 3.6p 450 = 3.6p _ _ 3.6p 450 3.6 3.6 125 = p = Given equation Substitution Property of Equality Simplify. Division Property of Equality Simplify
. p = 125 pixels Symmetric Property of Equality Look Back Check your answer by substituting it back into the original formula. sr = 3.6p (75) (6) = 3.6 (125) 450 = 450 ✓ AB represents the ̶̶ length of AB, so you can think of AB as a variable representing a number. 2. What is the temperature in degrees Celsius C when it is 86°F? Solve the equation C = 5 _ (F - 32) for C and justify each step. 9 Like algebra, geometry also uses numbers, variables, and operations. For example, segment lengths and angle measures are numbers. So you can use these same properties of equality to write algebraic proofs in geometry. 2- 5 Algebraic Proof 105 105 1234�� E X A M P L E 3 Solving an Equation in Geometry Write a justification for each step. KM = KL + LM 5x - 4 = (x + 3) + (2x - 1) 5x - 4 = 3x + 2 2x - 4 = 2 2x = 6 x = 3 Segment Addition Postulate Substitution Property of Equality Simplify. Subtraction Property of Equality Addition Property of Equality Division Property of Equality 3. Write a justification for each step. m∠ABC = m∠ABD + m∠DBC 8x ° = (3x + 5) ° + (6x - 16) ° 8x = 9x - 11 -x = -11 x = 11 You learned in Chapter 1 that segments with equal lengths are congruent and that angles with equal measures are congruent. So the Reflexive, Symmetric, and Transitive Properties of Equality have corresponding properties of congruence. Properties of Congruence SYMBOLS EXAMPLE Reflexive Property of Congruence figure A ≅ figure A (Reflex. Prop. of ≅) Symmetric Property of Congruence ̶̶ EF ≅ ̶̶ EF If figure A ≅ figure B, then figure B ≅ figure A. (Sym. Prop. of ≅) If ∠1 ≅ ∠2, then ∠2 ≅ ∠1. Transitive Property of Congruence If figure A ≅ figure B and figure B ≅ figure C, then figure A ≅ figure C. (Trans. Prop. of ≅) ̶̶ PQ ≅
̶̶ PQ ≅ ̶̶ RS and ̶̶ TU. If then ̶̶ RS ≅ ̶̶ TU, E X A M P L E 4 Identifying Properties of Equality and Congruence Identify the property that justifies each statement. Numbers are equal (=) and figures are congruent (≅). A m∠1 = m∠1 B ̶̶ XY ≅ ̶̶ VW, so ̶̶ VW ≅ ̶̶ XY. C ∠ABC ≅ ∠ABC Reflex. Prop. of = Sym. Prop. of ≅ Reflex. Prop. of ≅ D ∠1 ≅ ∠2, and ∠2 ≅ ∠3. So ∠1 ≅ ∠3. Trans. Prop. of ≅ Identify the property that justifies each statement. 4a. DE = GH, so GH = DE. 4c. 0 = a, and a = x. So 0 = x. 4b. 94° = 94° 4d. ∠A ≅ ∠Y, so ∠Y ≅ ∠A. 106 106 Chapter 2 Geometric Reasoning �� THINK AND DISCUSS 1. Tell what property you would use to solve the equation k _ 6 2. Explain when to use a congruence symbol instead of an equal sign. = 3.5. 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, write an example of the property, using the correct symbol. 2-5 Exercises Exercises KEYWORD: MG7 2-5 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Write the definition of proof in your own words Multi-Step Solve each equation. Write a justification for each step. p. 104 2. y + 1 = 5 3. t - 3.2 = -8.3 4. 2p - 30 = -4p + 6 n = 3 _ 6. 1 _ 4 2 5. x + 3 _ -2 = 8 7. 0 = 2 (r - 3. Nutrition Amy’s favorite breakfast cereal has 102 Calories per serving. The equation p. 105 C = 9f + 90 relates the grams of fat f in one serving to the Calories C in one serving. How many grams of fat are in one serving of the
cereal? Solve the equation for f and justify each step. 9. Movie Rentals The equation C = $5.75 + $0.89m relates the number of movie rentals m to the monthly cost C of a movie club membership. How many movies did Elias rent this month if his membership cost $11.98? Solve the equation for m and justify each step Write a justification for each step. p. 106 10. 11. AB = BC 5y + 6 = 2y + 21 3y + 6 = 21 3y = 15 y = 5 PQ + QR = PR 3n + 25 = 9n -5 25 = 6n -5 30 = 6n. 106 12. ̶̶ AB ≅ ̶̶ AB Identify the property that justifies each statement. 14. x = y, so y = x. 13. m∠1 = m∠2, and m∠2 = m∠4. So m∠1 = m∠4. ̶̶ PR. So ̶̶ YZ, and ̶̶ YZ ≅ ̶̶ ST ≅ ̶̶ ST ≅ ̶̶ PR. 15. 2- 5 Algebraic Proof 107 107 �� Independent Practice Multi-Step Solve each equation. Write a justification for each step. PRACTICE AND PROBLEM SOLVING For See Exercises Example 16. 5x - 3 = 4 (x + 2) 17. 1.6 = 3.2n 19. - (h + 3) = 72 20. 9y + 17 = -19 - 2 = -10 18. z _ 3 21. 1 _ (p - 16) = 13 2 16–21 22 23–24 25–28 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S7 Application Practice p. S29 22. Ecology The equation T = 0.03c + 0.05b relates the numbers of cans c and bottles b collected in a recycling rally to the total dollars T raised. How many cans were collected if $147 was raised and 150 bottles were collected? Solve the equation for c and justify each step. Write a justification for each step. 23. m∠XYZ = m∠2 + m∠3 4n - 6 = 58 + (2n - 12
) 4n - 6 = 2n + 46 2n - 6 = 46 2n = 52 n = 26 24. m∠WYV = m∠1 + m∠2 5n = 3 (n - 2) + 58 5n = 3n - 6 + 58 5n = 3n + 52 2n = 52 n = 26 Identify the property that justifies each statement. 25. ̶̶ KL ≅ ̶̶ PR, so ̶̶ PR ≅ ̶̶ KL. 26. 412 = 412 27. If a = b and b = 0, then a = 0. 28. figure A ≅ figure A 29. Estimation Round the numbers in the equation 2 (3.1x - 0.87) = 94.36 to the nearest whole number and estimate the solution. Then solve the equation, justifying each step. Compare your estimate to the exact solution. Use the indicated property to complete each statement. 30. Reflexive Property of Equality: 3x - 1 =? ̶̶̶ 31. Transitive Property of Congruence: If ∠ A ≅ ∠ X and ∠ X ≅ ∠T, then 32. Symmetric Property of Congruence: If ̶̶ BC ≅ ̶̶ NP, then?. ̶̶̶?. ̶̶̶ 33. Recreation The north campground is midway between the Northpoint Overlook and the waterfall. Use the midpoint formula to find the values of x and y, and justify each step. 34. Business A computer repair technician charges $35 for each job plus $21 per hour of labor and 110% of the cost of parts. The total charge for a 3-hour job was $169.50. What was the cost of parts for this job? Write and solve an equation and justify each step in the solution. 35. Finance Morgan spent a total of $1,733.65 on her car last year. She spent $92.50 on registration, $79.96 on maintenance, and $983 on insurance. She spent the remaining money on gas. She drove a total of 10,820 miles. a. How much on average did the gas cost per mile? Write and solve an equation and justify each step in the solution. b. What if…? Suppose Morgan’s car averages 32 miles per gallon of gas. How much on average did Morgan pay for a gallon of gas? 36. Critical
Thinking Use the definition of segment congruence and the properties of equality to show that all three properties of congruence are true for segments. 108 108 Chapter 2 Geometric Reasoning (1, y)NorthpointOverlook Northcampground(3, 5)(x, 1)WaterfallFinal file 2/25/05Campground mapHolt Rinehart WinstonGeometry 2007 Texasge07sec02l05003a Karen Minot(415)883-6560�� 37. This problem will prepare you for the Multi-Step TAKS Prep on page 126. Recall from Algebra 1 that the Multiplication and Division Properties of Inequality tell you to reverse the inequality sign when multiplying or dividing by a negative number. a. Solve the inequality x + 15 ≤ 63 and write a justification for each step. b. Solve the inequality -2x > 36 and write a justification for each step. 38. Write About It Compare the conclusion of a deductive proof and a conjecture based on inductive reasoning. 39. Which could NOT be used to justify the statement ̶̶ AB ≅ ̶̶ CD? Definition of congruence Symmetric Property of Congruence Reflexive Property of Congruence Transitive Property of Congruence 40. A club membership costs $35 plus $3 each time t the member uses the pool. Which equation represents the total cost C of the membership? C + 35 = 3t C = 35 + 3t 35 = C + 3t 41. Which statement is true by the Reflexive Property of Equality? ̶̶ CD = ̶̶ RT ≅ x = 35 ̶̶ CD ̶̶ TR C = 35t + 3 CD = CD 42. Gridded Response In the triangle, m∠1 + m∠2 + m∠3 = 180°. If m∠3 = 2m∠1 and m∠1 = m∠2, find m∠3 in degrees. CHALLENGE AND EXTEND 43. In the gate, PA = QB, QB = RA, and PA = 18 in. Find PR, and justify each step. 44. Critical Thinking Explain why there is no Addition Property of Congruence. 45. Algebra Justify each step in the solution of the inequality 7 - 3x > 19. SPIRAL REVIEW 46. The members of a high
school band have saved $600 for a trip. They deposit the money in a savings account. What additional information is needed to find the amount of interest the account earns during a 3-month period? (Previous course) Use a compass and straightedge to construct each of the following. (Lesson 1-2) 47. ̶̶ JK congruent to ̶̶̶ MN 48. a segment bisector of ̶̶ JK Identify whether each conclusion uses inductive or deductive reasoning. (Lesson 2-3) 49. A triangle is obtuse if one of its angles is obtuse. Jacob draws a triangle with two acute angles and one obtuse angle. He concludes that the triangle is obtuse. 50. Tonya studied 3 hours for each of her last two geometry tests. She got an A on both tests. She concludes that she will get an A on the next test if she studies for 3 hours. 2- 5 Algebraic Proof 109 109 �� 2-6 Geometric Proof TEKS G.1.A Geometric structure: develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems. Also G.3.B, G.3.C, G.3.E Objectives Write two-column proofs. Prove geometric theorems by using deductive reasoning. Vocabulary theorem two-column proof Who uses this? To persuade your parents to increase your allowance, your argument must be presented logically and precisely. When writing a geometric proof, you use deductive reasoning to create a chain of logical steps that move from the hypothesis to the conclusion of the conjecture you are proving. By proving that the conclusion is true, you have proven that the original conjecture is true When writing a proof, it is important to justify each logical step with a reason. You can use symbols and abbreviations, but they must be clear enough so that anyone who reads your proof will understand them. E X A M P L E 1 Writing Justifications When a justification is based on more than the previous step, you can note this after the reason, as in Example 1 Step 5. Write a justification for each step, given that ∠ A and ∠B are complementary and ∠ A ≅ ∠C. 1. ∠ A and ∠B are complementary. 2. m∠ A + m∠B = 90° 3. ∠
A ≅ ∠C 4. m∠ A = m∠C 5. m∠C + m∠B = 90° 6. ∠C and ∠B are complementary. Given information Def. of comp.  Given information Def. of ≅  Subst. Prop. of = Def. of comp.  Steps 2, 4 1. Write a justification for each step, given that B is the midpoint ̶̶ AC and ̶̶ AB ≅ ̶̶ EF. of 1. B is the midpoint of 2. 3. 4. ̶̶ AB ≅ ̶̶ AB ≅ ̶̶ BC ≅ ̶̶ BC ̶̶ EF ̶̶ EF ̶̶ AC. A theorem is any statement that you can prove. Once you have proven a theorem, you can use it as a reason in later proofs. Theorem THEOREM HYPOTHESIS CONCLUSION 2-6-1 Linear Pair Theorem If two angles form a linear pair, then they are supplementary. ∠A and ∠B form a linear pair. ∠A and ∠B are supplementary. 110 110 Chapter 2 Geometric Reasoning �� Theorem THEOREM HYPOTHESIS CONCLUSION 2-6-2 Congruent Supplements Theorem If two angles are supplementary to the same angle (or to two congruent angles), then the two angles are congruent. ∠1 and ∠2 are supplementary. ∠2 and ∠3 are supplementary. ∠1 ≅ ∠3 A geometric proof begins with Given and Prove statements, which restate the hypothesis and conclusion of the conjecture. In a two-column proof, you list the steps of the proof in the left column. You write the matching reason for each step in the right column. E X A M P L E 2 Completing a Two-Column Proof Fill in the blanks to complete a two-column proof of the Linear Pair Theorem. Given: ∠1 and ∠2 form a linear pair. Prove: ∠1 and ∠2 are supplementary. Proof: Statements Reasons 1. ∠1 and ∠2 form a linear pair. 1. Given 2.  BA and �
� BC form a line. 3. m∠ABC = 180° 4. a. 5. b.? ̶̶̶̶̶̶? ̶̶̶̶̶̶ 6. ∠1 and ∠2 are supplementary. 2. Def. of lin. pair 3. Def. of straight ∠ 4. ∠ Add. Post. 5. Subst. Steps 3, 4 6. c.? ̶̶̶̶̶̶ Since there is no other substitution property, the Substitution Property of Equality is often written as “Substitution” or “Subst.” Use the existing statements and reasons in the proof to fill in the blanks. a. m∠1 + m∠2 = m∠ABC b. m∠1 + m∠2 = 180° c. Def. of supp.  The ∠ Add. Post. is given as the reason. Substitute 180° for m∠ABC. The measures of supp.  add to 180° by def. 2. Fill in the blanks to complete a two-column proof of one case of the Congruent Supplements Theorem. Given: ∠1 and ∠2 are supplementary, and ∠2 and ∠3 are supplementary. Prove: ∠1 ≅ ∠3 Proof: Statements Reasons 1. a.? ̶̶̶̶̶̶ 2. m∠1 + m∠2 = 180° m∠2 + m∠3 = 180° 3. b.? ̶̶̶̶̶̶ 4. m∠2 = m∠2 5. m∠1 = m∠3 6. d.? ̶̶̶̶̶̶ 1. Given 2. Def. of supp.  3. Subst. 4. Reflex. Prop. of = 5. c.? ̶̶̶̶̶̶ 6. Def. of ≅  2- 6 Geometric Proof 111 111 �� Before you start writing a proof, you should plan out your logic. Sometimes you will be given a plan for a more challenging proof. This plan will detail the major steps of the proof for you. Theorems THEOREM HYPOTHESIS CONCLUSION 2-6-3 Right Angle Congruence Theorem All right angles are congruent. 2-6-4 Congruent Comple
ments Theorem If two angles are complementary to the same angle (or to two congruent angles), then the two angles are congruent. ∠A and ∠B are right angles. ∠A ≅ ∠B ∠1 and ∠2 are complementary. ∠2 and ∠3 are complementary. ∠1 ≅ ∠3 E X A M P L E 3 Writing a Two-Column Proof from a Plan If a diagram for a proof is not provided, draw your own and mark the given information on it. But do not mark the information in the Prove statement on it. Use the given plan to write a two-column proof of the Right Angle Congruence Theorem. Given: ∠1 and ∠2 are right angles. Prove: ∠1 ≅ ∠2 Plan: Use the definition of a right angle to write the measure of each angle. Then use the Transitive Property and the definition of congruent angles. Proof: Statements Reasons 1. ∠1 and ∠2 are right angles. 1. Given 2. m∠1 = 90°, m∠2 = 90° 3. m∠1 = m∠2 4. ∠1 ≅ ∠2 2. Def. of rt. ∠ 3. Trans. Prop. of = 4. Def. of ≅  3. Use the given plan to write a two-column proof of one case of the Congruent Complements Theorem. Given: ∠1 and ∠2 are complementary, and ∠2 and ∠3 are complementary. Prove: ∠1 ≅ ∠3 Plan: The measures of complementary angles add to 90° by definition. Use substitution to show that the sums of both pairs are equal. Use the Subtraction Property and the definition of congruent angles to conclude that ∠1 ≅ ∠3. The Proof Process 1. Write the conjecture to be proven. 2. Draw a diagram to represent the hypothesis of the conjecture. 3. State the given information and mark it on the diagram. 4. State the conclusion of the conjecture in terms of the diagram. 5. Plan your argument and prove the conjecture. 112 112 Chapter 2 Geometric Reasoning �� THINK AND DISCUSS 1. Which step in a proof should match the Prove statement? 2. Why is it important to include every logical step in
a proof? 3. List four things you can use to justify a step in a proof. 4. GET ORGANIZED Copy and complete the graphic organizer. In each box, describe the steps of the proof process. 2-6 Exercises Exercises KEYWORD: MG7 2-6 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. In a two-column proof, you list the? in the left column and the ̶̶̶̶ the right column. (statements or reasons)? in ̶̶̶̶. Write a justification for each step, given that m∠A = 60° and m∠B = 2m∠A. 2. A? is a statement you can prove. (postulate or theorem) ̶̶̶̶ p. 110 1. m∠A = 60°, m∠B = 2m∠A 2. m∠B = 2 (60°) 3. m∠B = 120° 4. m∠A + m∠ B = 60° + 120° 5. m∠A + m∠B = 180° 6. ∠A and ∠B are supplementary. Fill in the blanks to complete the two-column proof. p. 111 Given: ∠2 ≅ ∠3 Prove: ∠1 and ∠3 are supplementary. Proof: Statements Reasons 1. ∠2 ≅ ∠3 2. m∠2 = m∠3 3. b.? ̶̶̶̶̶̶ 1. Given 2. a.? ̶̶̶̶̶̶ 3. Lin. Pair Thm. 4. m∠1 + m∠2 = 180° 4. Def. of supp.  5. m∠1 + m∠3 = 180° 6. d.? ̶̶̶̶̶̶ 5. c.? Steps 2, 4 ̶̶̶̶̶ 6. Def. of supp. 112 5. Use the given plan to write a two-column proof. Given: X is the midpoint of ̶̶ YB Prove: ̶̶ AX ≅ ̶̶ AY, and Y is the midpoint of ̶̶ XB. Plan: By the definition of midpoint, Use the Transitive Property to conclude
that ̶̶ AX ≅ ̶̶ XY, and ̶̶ AX ≅ ̶̶ XY ≅ ̶̶ YB. ̶̶ YB. 2- 6 Geometric Proof 113 113 �� Independent Practice For See Exercises Example 6 7–8 9–10 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S7 Application Practice p. S29 PRACTICE AND PROBLEM SOLVING 6. Write a justification for each step, given that  BX bisects ∠ABC and m∠XBC = 45°. 1.  BX bisects ∠ABC. 2. ∠ABX ≅ ∠XBC 3. m∠ABX = m∠XBC 4. m∠XBC = 45° 5. m∠ABX = 45° 6. m∠ABX + m∠XBC = m∠ABC 7. 45° + 45° = m∠ABC 8. 90° = m∠ABC 9. ∠ABC is a right angle. Fill in the blanks to complete each two-column proof. 7. Given: ∠1 and∠2 are supplementary, and ∠3 and ∠4 are supplementary. ∠2 ≅ ∠3 Prove: ∠1 ≅ ∠4 Proof: Statements Reasons 1. ∠1 and ∠2 are supplementary. ∠3 and ∠4 are supplementary. 1. Given 2. a.? ̶̶̶̶̶ 3. m∠1 + m∠2 = m∠3 + m∠4 4. ∠2 ≅ ∠3 5. m∠2 = m∠3 6. c.? ̶̶̶̶̶ 7. ∠1 ≅ ∠4 2. Def. of supp.  3. b.? ̶̶̶̶̶ 4. Given 5. Def. of ≅  6. Subtr. Prop. of = Steps 3, 5 7. d.? ̶̶̶̶̶ 8. Given: ∠BAC is a right angle. ∠2 ≅ ∠3 Prove: ∠1 and ∠3 are complementary. Proof: Statements Reasons 1. �
�BAC is a right angle. 1. Given 2. m∠BAC = 90° 3. b.? ̶̶̶̶̶ 2. a.? ̶̶̶̶̶ 3. ∠ Add. Post. 4. m∠1 + m∠2 = 90° 4. Subst. Steps 2, 3 5. ∠2 ≅ ∠3 6. c.? ̶̶̶̶̶ 7. m∠1 + m∠3 = 90° 8. e.? ̶̶̶̶̶ 5. Given 6. Def. of ≅  7. d.? Steps 4, 6 ̶̶̶̶̶ 8. Def. of comp.  Use the given plan to write a two-column proof. 9. Given: Prove: ̶̶ BE ≅ ̶̶ AB ≅ ̶̶ CE, ̶̶ CD ̶̶ DE ≅ ̶̶ AE Plan: Use the definition of congruent segments to write the given information in terms of lengths. Then use the Segment Addition Postulate to show that AB = CD and thus ̶̶ AB ≅ ̶̶ CD. 114 114 Chapter 2 Geometric Reasoning �� Use the given plan to write a two-column proof. 10. Given: ∠1 and ∠3 are complementary, and ∠2 and ∠4 are complementary. ∠3 ≅ ∠4 Prove: ∠1 ≅ ∠2 Plan: Since ∠1 and ∠3 are complementary and ∠2 and ∠4 are complementary, both pairs of angle measures add to 90°. Use substitution to show that the sums of both pairs are equal. Since ∠3 ≅ ∠4, their measures are equal. Use the Subtraction Property of Equality and the definition of congruent angles to conclude that ∠1 ≅ ∠2. Engineering Find each angle measure. Find each angle measure. 11. 11. m∠1 12. m∠2 13. m∠3 The Bluff Dale Bridge, constructed by Texas builder William Flinn in 1891, is the oldest known cable-stayed bridge in the United States. 14. Engineering The Bluff Dale Bridge is 140 feet long and spans the Paluxy River in Bluff Dale, Texas. If ∠1 ≅ ∠2, which
theorem can you use to conclude that ∠3 ≅ ∠4? 15. Critical Thinking Explain why there are two cases to consider when proving the Congruent Supplements Theorem and the Congruent Complements Theorem. Tell whether each statement is sometimes, always, or never true. 16. An angle and its complement are congruent. 17. A pair of right angles forms a linear pair. 18. An angle and its complement form a right angle. 19. A linear pair of angles is complementary. Algebra Find the value of each variable. 20. 21. 22. � � � � 23. Write About It How are a theorem and a postulate alike? How are they different? 24. This problem will prepare you for the Multi-Step TAKS Prep on page 126. Sometimes you may be asked to write a proof without a specific statement of the Given and Prove information being provided for you. For each of the following situations, use the triangle to write a Given and Prove statement. a. The segment connecting the midpoints of two sides of a triangle is half as long as the third side. b. The acute angles of a right triangle are complementary. c. In a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse. 2- 6 Geometric Proof 115 115 �� 25. Which theorem justifies the conclusion that ∠1 ≅ ∠4? Linear Pair Theorem Congruent Supplements Theorem Congruent Complements Theorem Right Angle Congruence Theorem 26. What can be concluded from the statement m∠1 + m∠2 = 180°? ∠1 and ∠2 are congruent. ∠1 and ∠2 are supplementary. ∠1 and ∠2 are complementary. ∠1 and ∠2 form a linear pair. 27. Given: Two angles are complementary. The measure of one angle is 10° less than the measure of the other angle. Conclusion: The measures of the angles are 85° and 95°. Which statement is true? The conclusion is correct because 85° is 10° less than 95°. The conclusion is verified by the first statement given. The conclusion is invalid because the angles are not congruent. The conclusion is contradicted by the first statement given. CHALLENGE AND EXTEND 28. Write a
two-column proof. Given: m∠LAN = 30°, m∠1 = 15° Prove:  AM bisects ∠LAN. Multi-Step Find the value of the variable and the measure of each angle. 29. 30. SPIRAL REVIEW The table shows the number of tires replaced by a repair company during one week, classified by the mileage on the tires when they were replaced. Use the table for Exercises 31 and 32. (Previous course) 31. What percent of the tires had mileage between 40,000 and 49,999 when replaced? 32. If the company replaces twice as many tires next week, about how many tires would you expect to have lasted between 80,000 and 89,999 miles? Mileage on Replaced Tires Mileage Tires 40,000–49,999 50,000–59,999 60,000–69,999 70,000–79,999 80,000–89,999 60 82 54 40 14 Sketch a figure that shows each of the following. (Lesson 1-1) 33. Through any two collinear points, there is more than one plane containing them. 34. A pair of opposite rays forms a line. Identify the property that justifies each statement. (Lesson 2-5) 35. ̶̶ JK ≅ ̶̶ KL, so ̶̶ KL ≅ ̶̶ JK. 36. If m = n and n = p, then m = p. 116 116 Chapter 2 Geometric Reasoning �� 2-6 Use with Lesson 2-6 Activity Design Plans for Proofs Sometimes the most challenging part of writing a proof is planning the logical steps that will take you from the Given statement to the Prove statement. Like working a jigsaw puzzle, you can start with any piece. Write down everything you know from the Given statement. If you don’t see the connection right away, start with the Prove statement and work backward. Then connect the pieces into a logical order. TEKS G.1.A Geometric structure: develop an awareness of the structure of a mathematical system... Also G.3.B, G.3.C, G.3.E Prove the Common Angles Theorem. Given: ∠AXB ≅ ∠CXD Prove: ∠AXC
≅ ∠BXD 1 Start by considering the difference in the Given and Prove statements. How does ∠AXB compare to ∠AXC? How does ∠CXD compare to ∠BXD? In both cases, ∠BXC is combined with the first angle to get the second angle. 2 The situation involves combining adjacent angle measures, so list any definitions, properties, postulates, and theorems that might be helpful. Definition of congruent angles, Angle Addition Postulate, properties of equality, and Reflexive, Symmetric, and Transitive Properties of Congruence 3 Start with what you are given and what you are trying to prove and then work toward the middle. ∠AXB ≅ ∠CXD m∠AXB = m∠CXD??? m∠AXC = m∠BXD ∠AXC ≅ ∠BXD The first reason will be “Given.” Def. of ≅ ??? Def. of ≅  The last statement will be the Prove statement. 4 Based on Step 1, ∠BXC is the missing piece in the middle of the logical flow. So write down what you know about ∠BXC. ∠BXC ≅ ∠BXC m∠BXC = m∠BXC Reflex. Prop. of ≅ Reflex. Prop. of = 5 Now you can see that the Angle Addition Postulate needs to be used to complete the proof. m∠AXB + m∠BXC = m∠AXC m∠BXC + m∠CXD = m∠BXD ∠ Add. Post. ∠ Add. Post. 6 Reorder the pieces above to write a two-column proof of the Common Angles Theorem. Try This 1. Describe how a plan for a proof differs from the actual proof. 2. Write a plan and a two-column proof. BD bisects ∠ABC. Given: Prove: 2m∠1 = m∠ABC 3. Write a plan and a two-column proof. Given: ∠LXN is a right angle. Prove: ∠1 and ∠2 are complementary. 2-6 Geometry Lab 117 117 ��
�� 2-7 Flowchart and Paragraph Proofs TEKS G.1.A Geometric structure: develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems. Also G.2.B, G.3.C, G.3.E Objectives Write flowchart and paragraph proofs. Prove geometric theorems by using deductive reasoning. Vocabulary flowchart proof paragraph proof Why learn this? Flowcharts make it easy to see how the steps of a process are linked together. A second style of proof is a flowchart proof, which uses boxes and arrows to show the structure of the proof. The steps in a flowchart proof move from left to right or from top to bottom, shown by the arrows connecting each box. The justification for each step is written below the box. Theorem 2-7-1 Common Segments Theorem THEOREM HYPOTHESIS CONCLUSION Given collinear points A, B, C, and D arranged ̶̶ CD, then as shown, if ̶̶ AC ≅ ̶̶ AB ≅ ̶̶ BD. ̶̶ AB ≅ ̶̶ CD ̶̶ AC ≅ ̶̶ BD E X A M P L E 1 Reading a Flowchart Proof Use the given flowchart proof to write a two-column proof of the Common Segments Theorem. Given: Prove: ̶̶ AB ≅ ̶̶ AC ≅ ̶̶ CD ̶̶ BD Flowchart proof: �� Two-column proof: Statements Reasons ̶̶ AB ≅ ̶̶ CD 1. 2. AB = CD 3. BC = BC 4. AB + BC = BC + CD 1. Given 2. Def. of ≅ segs. 3. Reflex. Prop. of = 4. Add. Prop. of = 5. AB + BC = AC, BC + CD = BD 5. Seg. Add. Post. 6. AC = BD ̶̶ BD ̶̶ AC ≅ 7. 6. Subst. 7. Def. of ≅ segs. 118 118 Chapter 2 Geometric Reasoning �� 1. Use the given flowchart
proof to write a two-column proof. Given: RS = UV, ST = TU Prove: ̶̶ RT ≅ ̶̶ TV Flowchart proof: E X A M P L E 2 Writing a Flowchart Proof Use the given two-column proof to write a flowchart proof of the Converse of the Common Segments Theorem. Given: Prove: ̶̶ AC ≅ ̶̶ AB ≅ ̶̶ BD ̶̶ CD Two-column proof: Statements Reasons Like the converse of a conditional statement, the converse of a theorem is found by switching the hypothesis and conclusion. ̶̶ AC ≅ ̶̶ BD 1. 2. AC = BD 1. Given 2. Def. of ≅ segs. 3. AB + BC = AC, BC + CD = BD 3. Seg. Add. Post. 4. AB + BC = BC + CD 5. BC = BC 6. AB = CD ̶̶ CD ̶̶ AB ≅ 7. 4. Subst. Steps 2, 3 5. Reflex. Prop. of = 6. Subtr. Prop. of = 7. Def. of ≅ segs. Flowchart proof: 2. Use the given two-column proof to write a flowchart proof. Given: ∠2 ≅ ∠4 Prove: m∠1 = m∠3 Two-column proof: Statements Reasons 1. ∠2 ≅ ∠4 2. ∠1 and ∠2 are supplementary. ∠3 and ∠4 are supplementary. 3. ∠1 ≅ ∠3 4. m∠1 = m∠3 1. Given 2. Lin. Pair Thm. 3. ≅ Supps. Thm. 4. Def. of ≅  2-7 Flowchart and Paragraph Proofs 119 119 �� A paragraph proof is a style of proof that presents the steps of the proof and their matching reasons as sentences in a paragraph. Although this style of proof is less formal than a two-column proof, you still
must include every step. Theorems THEOREM HYPOTHESIS CONCLUSION 2-7-2 Vertical Angles Theorem Vertical angles are congruent. ∠A and ∠B are vertical angles. ∠A ≅ ∠B 2-7-3 If two congruent angles are supplementary, then each angle is a right angle. (≅  supp. → rt. ) ∠1 ≅ ∠2 ∠1 and ∠2 are supplementary. ∠1 and ∠2 are right angles. E X A M P L E 3 Reading a Paragraph Proof Use the given paragraph proof to write a two-column proof of the Vertical Angles Theorem. Given: ∠1 and ∠3 are vertical angles. Prove: ∠1 ≅ ∠3 � � � Paragraph proof: ∠1 and ∠3 are vertical angles, so they are formed by intersecting lines. Therefore ∠1 and ∠2 are a linear pair, and ∠2 and ∠3 are a linear pair. By the Linear Pair Theorem, ∠1 and ∠2 �� are supplementary, and ∠2 and ∠3 are supplementary. So by the �� Congruent Supplements Theorem, ∠1 ≅ ∠3. �� Two-column proof: Statements Reasons 1. ∠1 and ∠3 are vertical angles. 1. Given 2. ∠1 and ∠3 are formed by intersecting lines. 2. Def. of vert.  3. ∠1 and ∠2 are a linear pair. ∠2 and ∠3 are a linear pair. 4. ∠1 and ∠2 are supplementary. ∠2 and ∠3 are supplementary. 5. ∠1 ≅ ∠3 3. Def. of lin. pair 4. Lin. Pair Thm. 5. ≅ Supps. Thm. 3. Use the given paragraph proof to write a two-column proof. Given: ∠WXY is a right angle. ∠1 ≅ ∠3 Prove: ∠1 and ∠2 are complementary. Paragraph proof: Since ∠WXY is a right angle, m∠WXY = 90° by the definition of a right angle. By the Angle Addition Postulate, m∠WXY
= m∠2 + m∠3. By substitution, m∠2 + m∠3 = 90°. Since ∠1 ≅ ∠3, m∠1 = m∠3 by the definition of congruent angles. Using substitution, m∠2 + m∠1 = 90°. Thus by the definition of complementary angles, ∠1 and ∠2 are complementary. 120 120 Chapter 2 Geometric Reasoning �� Writing a Proof When I have to write a proof and I don’t see how to start, I look at what I’m supposed to be proving and see if it makes sense. If it does, I ask myself why. Sometimes this helps me to see what the reasons in the proof might be. If all else fails, I just start writing down everything I know based on the diagram and the given statement. By brainstorming like this, I can usually figure out the steps of the proof. You can even write each thing on a separate piece of paper and arrange the pieces of paper like a flowchart. Claire Jeffords Riverbend High School E X A M P L E 4 Writing a Paragraph Proof Use the given two-column proof to write a paragraph proof of Theorem 2-7-3. Given: ∠1 and ∠2 are supplementary. ∠1 ≅ ∠2 Prove: ∠1 and ∠2 are right angles. Two-column proof: Statements Reasons 1. ∠1 and ∠2 are supplementary. 1. Given ∠1 ≅ ∠2 2. m∠1 + m∠2 = 180° 3. m∠1 = m∠2 4. m∠1 + m∠1 = 180° 5. 2m∠1 = 180° 6. m∠1 = 90° 7. m∠2 = 90° 2. Def. of supp.  3. Def. of ≅  Step 1 4. Subst. Steps 2, 3 5. Simplification 6. Div. Prop. of = 7. Trans. Prop. of = Steps 3, 6 8. ∠1 and ∠2 are right angles. 8. Def. of rt. ∠ Paragraph proof: ∠1 and ∠2 are supplementary, so m∠1 + m∠2 = 180° by the definition of supplementary angles. They are
also congruent, so their measures are equal by the definition of congruent angles. By substitution, m∠1 + m∠1 = 180°, so m∠1 = 90° by the Division Property of Equality. Because m∠1 = m∠2, m∠2 = 90° by the Transitive Property of Equality. So both are right angles by the definition of a right angle. 4. Use the given two-column proof to write a paragraph proof. Given: ∠1 ≅ ∠4 Prove: ∠2 ≅ ∠3 Two-column proof: Statements Reasons 1. ∠1 ≅ ∠4 1. Given 2. ∠1 ≅ ∠2, ∠3 ≅ ∠4 2. Vert.  Thm. 3. ∠2 ≅ ∠4 4. ∠2 ≅ ∠3 3. Trans. Prop. of ≅ Steps 1, 2 4. Trans. Prop. of ≅ Steps 2, 3 2-7 Flowchart and Paragraph Proofs 121 121 �� THINK AND DISCUSS 1. Explain why there might be more than one correct way to write a proof. 2. Describe the steps you take when writing a proof. 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, describe the proof style in your own words. 2-7 Exercises Exercises KEYWORD: MG7 2-7 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. In a? proof, the logical order is represented by arrows that connect each step. ̶̶̶̶ (flowchart or paragraph) 2. The steps and reasons of a (flowchart or paragraph)? proof are written out in sentences. ̶̶̶̶. Use the given flowchart proof to write p. 118 a two-column proof. Given: ∠1 ≅ ∠2 Prove: ∠1 and ∠2 are right angles. Flowchart proof. Use the given two-column proof to write p. 119 a flowchart proof. Given: ∠2 and ∠4 are supplementary. Prove: m∠2 = m∠3 Two-column proof: Statements Reasons 1. ∠2 and ∠4 are supplementary. 1. Given 2. �
�3 and ∠4 are supplementary. 2. Lin. Pair Thm. 3. ∠2 ≅ ∠3 4. m∠2 = m∠3 3. ≅ Supps. Thm. Steps 1, 2 4. Def. of ≅  122 122 Chapter 2 Geometric Reasoning ��. Use the given paragraph proof to write a two-column proof. p. 120 Given: ∠2 ≅ ∠4 Prove: ∠1 ≅ ∠3 Paragraph proof: By the Vertical Angles Theorem, ∠1 ≅ ∠2, and ∠3 ≅ ∠4. It is given that ∠2 ≅ ∠4. By the Transitive Property of Congruence, ∠1 ≅ ∠4, and thus ∠1 ≅ ∠3. Use the given two-column proof to write a paragraph proof. p. 121 Given: Prove:  BD bisects ∠ABC.  BG bisects ∠FBH. Two-column proof: Statements Reasons 1.  BD bisects ∠ABC. 1. Given 2. ∠1 ≅ ∠2 2. Def. of ∠ bisector 3. ∠1 ≅ ∠4, ∠2 ≅ ∠3 3. Vert.  Thm. 4. ∠4 ≅ ∠2 5. ∠4 ≅ ∠3 4. Trans. Prop. of ≅ Steps 2, 3 5. Trans. Prop. of ≅ Steps 3, 4 6.  BG bisects ∠FBH. 6. Def. of ∠ bisector Independent Practice For See Exercises Example 7 8 9 10 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S7 Application Practice p. S29 PRACTICE AND PROBLEM SOLVING 7. Use the given flowchart proof to write a two-column proof. Given: B is the midpoint of ̶̶ AC. AD = EC Prove: DB
= BE Flowchart proof: 8. Use the given two-column proof to write a flowchart proof. Given: ∠3 is a right angle. Prove: ∠4 is a right angle. Two-column proof: Statements Reasons 1. ∠3 is a right angle. 2. m∠3 = 90° 1. Given 2. Def. of rt. ∠ 3. ∠3 and ∠4 are supplementary. 3. Lin. Pair Thm. 4. m∠3 + m∠4 = 180° 5. 90° + m∠4 = 180° 6. m∠4 = 90° 7. ∠4 is a right angle. 4. Def. of supp.  5. Subst. Steps 2, 4 6. Subtr. Prop. of = 7. Def. of rt. ∠ 2-7 Flowchart and Paragraph Proofs 123 123 �� 9. Use the given paragraph proof to write a two-column proof. Given: ∠1 ≅ ∠4 Prove: ∠2 and ∠3 are supplementary. Paragraph proof: ∠4 and ∠3 form a linear pair, so they are supplementary by the Linear Pair Theorem. Therefore, m∠4 + m∠3 = 180°. Also, ∠1 and ∠2 are vertical angles, so ∠1 ≅ ∠2 by the Vertical Angles Theorem. It is given that ∠1 ≅ ∠4. So by the Transitive Property of Congruence, ∠4 ≅ ∠2, and by the definition of congruent angles, m∠4 = m∠2. By substitution, m∠2 + m∠3 = 180°, so ∠2 and ∠3 are supplementary by the definition of supplementary angles. 10. Use the given two-column proof to write a paragraph proof. Given: ∠1 and ∠2 are complementary. Prove: ∠2 and ∠3 are complementary. Two-column proof: Statements Reasons 1. ∠1 and ∠2 are complementary. 1. Given 2. m∠1 + m∠
2 = 90° 3. ∠1 ≅ ∠3 4. m∠1 = m∠3 5. m∠3 + m∠2 = 90° 2. Def. of comp.  3. Vert.  Thm. 4. Def. of ≅  5. Subst. Steps 2, 4 6. ∠2 and ∠3 are complementary. 6. Def. of comp.  Find each measure and name the theorem that justifies your answer. 11. AB 12. m∠2 13. m∠3 Algebra Find the value of each variable. 14. 15. 16. 17. /////ERROR ANALYSIS///// Below are two drawings for the given proof. Which is incorrect? Explain the error. ̶̶ BC Given: Prove: ∠A ≅ ∠C ̶̶ AB ≅ 18. This problem will prepare you for the Multi-Step TAKS Prep on page 126. Rearrange the pieces to create a flowchart proof. 124 124 Chapter 2 Geometric Reasoning ��x � ��x � ��y�� 19. Critical Thinking Two lines intersect, and one of the angles formed is a right angle. Explain why all four angles are congruent. 20. Write About It Which style of proof do you find easiest to write? to read? 21. Which pair of angles in the diagram must be congruent? ∠1 and ∠5 ∠3 and ∠4 ∠5 and ∠8 None of the above 22. What is the measure of ∠2? 38° 52° 128° 142° 23. Which statement is NOT true if ∠2 and ∠6 are supplementary? m∠2 + m∠6 = 180° ∠2 and ∠3 are supplementary. ∠1 and ∠6 are supplementary. m∠1 + m∠4 = 180° CHALLENGE AND EXTEND 24. Textiles Use the woven pattern to write a flowchart proof. Given: ∠1 ≅ ∠3 Prove: m∠4 + m∠5 = m�
�6 25. Write a two-column proof. Given: ∠AOC ≅ ∠BOD Prove: ∠AOB ≅ ∠COD 26. Write a paragraph proof. Given: ∠2 and ∠5 are right angles. m∠1 + m∠2 + m∠3 = m∠4 + m∠5 + m∠6 Prove: ∠1 ≅ ∠4 27. Multi-Step Find the value of each variable and the measures of all four angles. SPIRAL REVIEW Solve each system of equations. Check your solution. (Previous course) ⎧ 7x - y = -33 29. ⎨ 3x + y = -7 ⎩ ⎧ 28. ⎨ ⎩ y = -6x + 18 y = 2x + 14 ⎧ ⎨ ⎩ 30. 2x + y = 8 -x + 3y = 10 Use a protractor to draw an angle with each of the following measures. (Lesson 1-3) 31. 125° 32. 38° 33. 94° 34. 175° For each conditional, write the converse and a biconditional statement. (Lesson 2-4) 35. If a positive integer has more than two factors, then it is a composite number. 36. If a quadrilateral is a trapezoid, then it has exactly one pair of parallel sides. 2-7 Flowchart and Paragraph Proofs 125 125 �� SECTION 2B Mathematical Proof Intersection Inspection According to the U.S. Department of Transportation, it is ideal for two intersecting streets to form four 90° angles. If this is not possible, roadways should meet at an angle of 75° or greater for maximum safety and visibility. 1. Write a compound inequality to represent the range of measures an angle in an intersection should have. 2. Suppose that an angle in an intersection meets the guidelines specified by the U.S. Department of Transportation. Find the range of measures for the adjacent angle in the intersection. The intersection of West Elm Street and Lamar Boulevard has a history of car accidents. The Southland neighborhood association is circulating a petition to have the city reconstruct the intersection. A surveyor measured the intersection, and one of the angles measures
145°. 3. Given that m∠2 = 145°, write a two-column proof to show that m∠1 and m∠3 are less than 75°. 4. Write a paragraph proof to justify the argument that the intersection of West Elm Street and Lamar Boulevard should be reconstructed. 126 126 Chapter 2 Geometric Reasoning �� SECTION 2B Quiz for Lessons 2-5 Through 2-7 2-5 Algebraic Proof Solve each equation. Write a justification for each step. 1. m - 8 = 13 2. 4y - 1 = 27 3. - x _ 3 = 2 Identify the property that justifies each statement. 4. m∠XYZ = m∠PQR, so m∠PQR = m∠XYZ. ̶̶ AB ≅ ̶̶ AB 5. 6. ∠4 ≅ ∠A, and ∠A ≅ ∠1. So ∠4 ≅ ∠1. 7. k = 7, and m = 7. So k = m. 2-6 Geometric Proof 8. Fill in the blanks to complete the two-column proof. Given: m∠1 + m∠3 = 180° Prove: ∠1 ≅ ∠4 Proof: Statements Reasons 1. m∠1 + m∠3 = 180° 2. b.? ̶̶̶̶̶̶ 1. a.? ̶̶̶̶̶̶ 2. Def. of supp.  3. ∠3 and ∠4 are supplementary. 3. Lin. Pair Thm. 4. ∠3 ≅ ∠3 5. d.? ̶̶̶̶̶̶ 4. c.? ̶̶̶̶̶ 5. ≅ Supps. Thm. 9. Use the given plan to write a two-column proof of the Symmetric Property of ̶̶ EF ̶̶ AB Congruence. ̶̶ AB ≅ Given: ̶̶ EF ≅ Prove: Plan: Use the definition of congruent segments to write of equality. Then use the Symmetric Property of Equality to show that EF = AB. So ̶̶ AB by the definition of congruent segments. ̶̶ EF as a statement ̶̶ AB ≅ ̶̶ EF ≅ 2-7 Flowchart and Par
agraph Proofs Use the given two-column proof to write the following. Given: ∠1 ≅ ∠3 Prove: ∠2 ≅ ∠4 Proof: Statements Reasons 1. ∠1 ≅ ∠3 1. Given 2. ∠1 ≅ ∠2, ∠3 ≅ ∠4 2. Vert.  Thm. 3. ∠2 ≅ ∠3 4. ∠2 ≅ ∠4 3. Trans. Prop. of ≅ 4. Trans. Prop. of ≅ 10. a flowchart proof 11. a paragraph proof Ready to Go On? 127 127 �� EXTENSION EXTENSION Introduction to Symbolic Logic Objectives Analyze the truth value of conjunctions and disjunctions. Construct truth tables to determine the truth value of logical statements. Vocabulary compound statement conjunction disjunction truth table TEKS G.4.A Select an appropriate representation... in order to solve problems. Also G.3.C Symbolic logic is used by computer programmers, mathematicians, and philosophers to analyze the truth value of statements, independent of their actual meaning. A compound statement is created by combining two or more statements. Suppose p and q each represent a statement. Two compound statements can be formed by combining p and q: a conjunction and a disjunction. Compound Statements TERM WORDS SYMBOLS EXAMPLE Conjunction A compound statement that uses the word and p AND q p ⋀ q Pat is a band member AND Pat plays tennis. Disjunction A compound statement that uses the word or p OR q p ⋁ q Pat is a band member OR Pat plays tennis. A conjunction is true only when all of its parts are true. A disjunction is true if any one of its parts is true. E X A M P L E 1 Analyzing Truth Values of Conjunctions and Disjunctions Use p, q, and r to find the truth value of each compound statement. p: Washington, D.C., is the capital of the United States. q: The day after Monday is Tuesday. r: California is the largest state in the United States. A q ⋁ r B r ⋀ p Since q is true, the disjunction is true. Since r is false, the conjunction is false. Use the information given above to find the truth value of each compound statement. 1a. r ⋁ p
1b. p ⋀ q A table that lists all possible combinations of truth values for a statement is called a truth table. A truth table shows you the truth value of a compound statement, based on the possible truth values of its parts. Make sure you include all possible combinations of truth values for each piece of the compound statement 128 128 Chapter 2 Geometric Reasoning E X A M P L E 2 Constructing Truth Tables for Compound Statements Construct a truth table for the compound statement ∼u ⋀ (v ⋁ w). Since u, v, and w can each be either true or false, the truth table will have (2) (2) (2) = 8 rows. The negation (~) of a statement has the opposite truth valueu v ⋁ w ∼u ⋀ (v ⋁ w. Construct a truth table for the compound statement ∼u ⋀ ∼v. EXTENSION Exercises Exercises Use p, q, and r to find the truth value of each compound statement. p : The day after Friday is Sunday. q: 1 _ 2 r : If -4x - 2 = 10, then x = 3. = 0.5 1. r ⋀ q 4. q ⋀ ∼q 2. r ⋁ p 5. ∼q ⋁ q 3. p ⋁ r 6. q ⋁ r Construct a truth table for each compound statement. 7. s ⋀ ∼t 8. ∼u ⋁ t 9. ∼u ⋁ (s ⋀ t) Use a truth table to show that the two statements are logically equivalent. 10. p → q; ∼q → ∼p 11. q → p; ∼p → ∼q 12. A biconditional statement can be written as (p → q) ⋀ (q → p). Construct a truth table for this compound statement. 13. DeMorgan’s Laws state that ∼ (p ⋀ q) = ∼p ⋁ ∼q and that ∼ (p ⋁ q) = ∼p ⋀ ∼q. a. Use truth tables to show that both statements are true. b. If you think of disjunction and conjunction as inverse operations, DeMorgan’s Laws are similar to which algebraic property? 14. The Law of Disjunctive Inference states that if p ⋁ q
is true and p is false, then q must be true. a. Construct a truth table for p ⋁ q. b. Use the truth table to explain why the Law of Disjunctive Inference is true. Chapter 2 Extension 129 129 For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary biconditional statement...... 96 definition................... 97 paragraph proof............ 120 conclusion.................. 81 flowchart proof............. 118 polygon..................... 98 conditional statement........ 81 hypothesis.................. 81 proof...................... 104 conjecture.................. 74 inductive reasoning.......... 74 quadrilateral................ 98 contrapositive............... 83 inverse...................... 83 theorem................... 110 converse.................... 83 logically equivalent triangle..................... 98 counterexample............. 75 deductive reasoning......... 88 statements................ 83 negation.................... 82 truth value.................. 82 two-column proof..........
111 Complete the sentences below with vocabulary words from the list above. 1. A statement you can prove and then use as a reason in later proofs is a(n)?. ̶̶̶ 2.? is the process of using logic to draw conclusions from given facts, definitions, ̶̶̶ and properties. 3. A(n)? is a case in which a conjecture is not true. ̶̶̶ 4. A statement you believe to be true based on inductive reasoning is called a(n)?. ̶̶̶ 2-1 Using Inductive Reasoning to Make Conjectures (pp. 74–79) E X A M P L E S EXERCISES TEKS G.2.B, G.3.D, G.5.B ■ Find the next item in the pattern below. Make a conjecture about each pattern. Write the next two items. The red square moves in the counterclockwise direction. The next figure is. 5. ■ Complete the conjecture “The sum of two?.” ̶̶̶ odd numbers is List some examples and look for a pattern + 11 = 18 Complete each conjecture. 8. The sum of an even number and an odd number is?. ̶̶̶ The sum of two odd numbers is even. 9. The square of a natural number is?. ̶̶̶ ■ Show that the conjecture “For all non-zero integers, -x < x” is false by finding a counterexample. Pick positive and negative values for x and substitute to see if the conjecture holds. Let n = 3. Since -3 < 3, the conjecture holds. Let n = -5. Since - (-5) is 5 and 5 ≮ -5, the conjecture is false. n = -5 is a counterexample. 130 130 Chapter 2 Geometric Reasoning Determine if each conjecture is true. If not, write or draw a counterexample. 10. All whole numbers are natural numbers. ̶̶ BC. 11. If C is the midpoint of ̶̶ AB, then ̶̶ AC ≅ 12. If 2x + 3 = 15, then x = 6. 13. There are 28 days in February. 14. Draw a triangle. Construct the bisectors of each angle of the triangle. Make a conjecture about where the three angle bisectors intersect. �� 2-2 Conditional Statements (
pp. 81–87) TEKS G.3.A, G.3.C E X A M P L E S EXERCISES ■ Write a conditional statement from the sentence “A rectangle has congruent diagonals.” If a figure is a rectangle, then it has congruent diagonals. ■ Write the inverse, converse, and contrapositive of the conditional statement “If m∠1 = 35°, then ∠1 is acute.” Find the truth value of each. Converse: If ∠1 is acute, then m∠1 = 35°. Not all acute angles measure 35°, so this is false. Inverse: If m∠1 ≠ 35°, then ∠1 is not acute. You can draw an acute angle that does not measure 35°, so this is false. Contrapositive: If ∠1 is not acute, then m∠1 ≠ 35°. An angle that measures 35° must be acute. So this statement is true. Write a conditional statement from each Venn diagram. 15. 16. Determine if each conditional is true. If false, give a counterexample. 17. If two angles are adjacent, then they have a common ray. 18. If you multiply two irrational numbers, the product is irrational. Write the converse, inverse, and contrapositive of each conditional statement. Find the truth value of each. 19. If ∠X is a right angle, then m∠X = 90°. 20. If x is a whole number, then x = 2. 2-3 Using Deductive Reasoning to Verify Conjectures (pp. 88–93) TEKS G.2.B, E X A M P L E S EXERCISES G.3.B, G.3.C, G.3.E ■ Determine if the conjecture is valid by the Law of Detachment or the Law of Syllogism. Given: If 5c = 8y, then 2w = -15. If 5c = 8y, then x = 17. Conjecture: If 2w = -15, then x = 17. Let p be 5c = 8y, q be 2w = -15, and r be x = 17. Using symbols, the given information is written as p → q and p → r. Neither the Law of Detachment nor the Law
of Syllogism can be applied. The conjecture is not valid. Use the true statements below to determine whether each conclusion is true or false. Sue is a member of the swim team. When the team practices, Sue swims. The team begins practice when the pool opens. The pool opens at 8 A.M. on weekdays and at 12 noon on Saturday. 21. The swim team practices on weekdays only. 22. Sue swims on Saturdays. 23. Swim team practice starts at the same time every day. ■ Draw a conclusion from the given information. Given: If two points are distinct, then there is one line through them. A and B are distinct points. Let p be the hypothesis: two points are distinct. Let q be the conclusion: there is one line through the points. The statement “A and B are distinct points” matches the hypothesis, so you can conclude that there is one line through A and B. Use the following information for Exercises 24–26. The expression 2.15 + 0.07x gives the cost of a long-distance phone call, where x is the number of minutes after the first minute. If possible, draw a conclusion from the given information. If not possible, explain why. 24. The cost of Sara’s long-distance call is $2.57. 25. Paulo makes a long-distance call that lasts ten minutes. 26. Asa’s long-distance phone bill for the month is $19.05. Study Guide: Review 131 131 �� 2-4 Biconditional Statements and Definitions (pp. 96–101) TEKS G.3.A, G.3.B E X A M P L E S EXERCISES ■ For the conditional “If a number is divisible by 10, then it ends in 0”, write the converse and a biconditional statement. Converse: If a number ends in 0, then it is divisible by 10. Biconditional: A number is divisible by 10 if and only if it ends in 0. ■ Determine if the biconditional “The sides of a triangle measure 3, 7, and 15 if and only if the perimeter is 25” is true. If false, give a counterexample. Conditional: If the sides of a triangle measure 3, 7, and 15, then the perimeter is 25. True.
Converse: If the perimeter of a triangle is 25, then its sides measure 3, 7, and 15. False; a triangle with side lengths of 6, 10, and 9 also has a perimeter of 25. Therefore the biconditional is false. Determine if a true biconditional can be written from each conditional statement. If not, give a counterexample. 27. If 3 - 2x_ 5 = 2, then x = 5_. 2 28. If x < 0, then the value of x 4 is positive. 29. If a segment has endpoints at (1, 5) and (-3, 1), then its midpoint is (-1, 3). 30. If the measure of one angle of a triangle is 90°, then the triangle is a right triangle. Complete each statement to form a true biconditional. 31. Two angles are? if and only if the sum of ̶̶̶ their measures is 90°. 32. x 3 >0 if and only if x is?. ̶̶̶ 33. Trey can travel 100 miles in less than 2 hours if and only if his average speed is?. ̶̶̶ 34. The area of a square is equal to s 2 if and only if the perimeter of the square is?. ̶̶̶ 2-5 Algebraic Proof (pp. 104–109) TEKS G.3.B, G.3.C, G.3.E E X A M P L E S EXERCISES ■ Solve the equation 5x - 3 = -18. Write a justification for each step. 5x - 3 = -18 + 3 + 3 ̶̶̶ ̶̶̶̶̶ 5x = -15 -15_ 5x_ 5 5 x = -3 = Given Add. Prop. of = Simplify. Div. Prop. of = Simplify. ■ Write a justification for each step. RS = ST Given 5x - 18 = 4x x - 18 = 0 x = 18 Subst. Prop. of = Subtr. Prop. of = Add. Prop. of = Identify the property that justifies each statement. ■ ∠X ≅ ∠2, so ∠2 ≅ ∠X. Symmetric Property of Congruence ■ If m∠2 = 180° and m∠3 = 180°, then m∠2 = m∠
3. Transitive Property of Equality 132 132 Chapter 2 Geometric Reasoning Solve each equation. Write a justification for each step. 35. m_ -5 36. -47 = 3x - 59 + 3 = -4.5 Identify the property that justifies each statement. 37. a + b = a + b 38. If ∠RST ≅ ∠ABC, then ∠ABC ≅ ∠RST. 39. 2x = 9, and y = 9. So 2x = y. Use the indicated property to complete each statement. 40. Reflex. Prop. of ≅: figure ABCD ≅? ̶̶̶ 41. Sym. Prop. of =: If m∠2 = m∠5, then ̶̶ CD and ̶̶ AB ≅?. ̶̶̶ ̶̶ EF, 42. Trans. Prop. of ≅: If?. ̶̶̶ then ̶̶ AB ≅ 43. Kim borrowed money at an annual simple interest rate of 6% to buy a car. How much did she borrow if she paid $4200 in interest over the life of the 4-year loan? Solve the equation I = Prt for P and justify each step. �� 2-6 Geometric Proof (pp. 110–116) TEKS G.1.A, G.3.B, G.3.C, G.3.E E X A M P L E S EXERCISES ■ Write a justification for each step, given that m∠2 = 2m∠1. 1. ∠1 and ∠2 supp. 2. m∠1 + m∠2 = 180° 3. m∠2 = 2m∠1 4. m∠1 + 2m∠1 = 180° 5. 3m∠1 = 180° 6. m∠1 = 60° Lin. Pair Thm. Def. of supp.  Given Subst. Steps 2, 3 Simplify Div. Prop. of = ■ Use the given plan to write a two-column proof. Given: ̶̶ AD bisects ∠BAC. ∠1 ≅ ∠3 Prove: ∠2 ≅ ∠3 Plan: Use the definition of angle bisector to show that ∠1 ≅ ∠2. Use the Transitive Property to conclude that �
�2 ≅ ∠3. Two-column proof: Statements Reasons ̶̶̶ AD bisects ∠BAC. 1. 1. Given 2. ∠1 ≅ ∠2 3. ∠1 ≅ ∠3 4. ∠2 ≅ ∠3 2. Def. of ∠ bisector 3. Given 4. Trans. Prop. of ≅ 44. Write a justification for each step, given that ∠1 and ∠2 are complementary, and ∠1 ≅ ∠3. 1. ∠1 and ∠2 comp. 2. m∠1 + m∠2 = 90° 3. ∠1 ≅ ∠3 4. m∠1 = m∠3 5. m∠3 + m∠2 = 90° 6. ∠3 and ∠2 comp. 45. Fill in the blanks to complete the two-column proof. ̶̶ TU ≅ Given: Prove: SU + TU = SV Two-column proof: ̶̶ UV Statements Reasons ̶̶ TU ≅ ̶̶ UV 1. 2. b. 3. c.? ̶̶̶̶? ̶̶̶̶ 4. SU + TU = SV 1. a.? ̶̶̶̶ 2. Def. of ≅ segs. 3. Seg. Add. Post. 4. d.? ̶̶̶̶ Find the value of each variable. 46. 47. 2-7 Flowchart and Paragraph Proofs (pp. 118–125) TEKS G.1.A, G.2.B, G.3.C, G.3.E E X A M P L E S EXERCISES Use the two-column proof in the example for Lesson 2-6 above to write each of the following. ■ a flowchart proof ■ a paragraph proof ̶̶ AD bisects ∠BAC, ∠1 ≅ ∠2 by the Since definition of angle bisector. It is given that ∠1 ≅ ∠3. Therefore, ∠2 ≅ ∠3 by the Transitive Property of Congruence. Use the given plan to write each of the following. Given: ∠ADE and ∠DAE are complementary. ∠ADE and ∠BAC are complementary. Prove: ∠DAC �
� ∠BAE Plan: Use the Congruent Complements Theorem to show that ∠DAE ≅ ∠BAC. Since ∠CAE ≅ ∠CAE, ∠DAC ≅ ∠BAE by the Common Angles Theorem. 48. a flowchart proof 49. a paragraph proof Find the value of each variable and name the theorem that justifies your answer. 50. 51. Study Guide: Review 133 133 �� Find the next item in each pattern. 1. 2. 405, 135, 45, 15, … 3. Complete the conjecture “The sum of two even numbers is?. ” ̶̶̶ 4. Show that the conjecture “All complementary angles are adjacent” is false by finding a counterexample. 5. Identify the hypothesis and conclusion of the conditional statement “The show is cancelled if it rains.” 6. Write a conditional statement from the sentence “Parallel lines do not intersect.” Determine if each conditional is true. If false, give a counterexample. 7. If two lines intersect, then they form four right angles. 8. If a number is divisible by 10, then it is divisible by 5. Use the conditional “If you live in the United States, then you live in Kentucky” for Items 9–11. Write the indicated type of statement and determine its truth value. 9. converse 10. inverse 11. contrapositive 12. Determine if the following conjecture is valid by the Law of Detachment. Given: If it is colder than 50°F, Tom wears a sweater. It is 46°F today. Conjecture: Tom is wearing a sweater. 13. Use the Law of Syllogism to draw a conclusion from the given information. Given: If a figure is a square, then it is a quadrilateral. If a figure is a quadrilateral, then it is a polygon. Figure ABCD is a square. 14. Write the conditional statement and converse within the biconditional “Chad will work on Saturday if and only if he gets paid overtime.” 15. Determine if the biconditional “B is the midpoint of ̶̶ AC iff
AB = BC” is true. If false, give a counterexample. Solve each equation. Write a justification for each step. 16. 8 - 5s = 1 17. 0.4t + 3 = 1.6 18. 38 = -3w + 2 Identify the property that justifies each statement. 19. If 2x = y and y = 7, then 2x = 7. 21. ∠X ≅ ∠P, and ∠P ≅ ∠D. So ∠X ≅ ∠D. Use the given plan to write a proof in each format. 20. m∠DEF = m∠DEF ̶̶ XY ≅ ̶̶ XY, then ̶̶ ST ≅ 22. If ̶̶ ST.  FB bisects ∠AFC. Given: ∠AFB ≅ ∠EFD Prove: Plan: Since vertical angles are congruent, ∠EFD ≅ ∠BFC. Use the Transitive Property to conclude that ∠AFB ≅ ∠BFC. Thus  FB bisects ∠AFC by the definition of angle bisector. 23. two-column proof 24. paragraph proof 25. flowchart proof 134 134 Chapter 2 Geometric Reasoning �� FOCUS ON SAT MATHEMATICS SUBJECT TESTS Some colleges require that you take the SAT Subject Tests. There are two math subject tests—Level 1 and Level 2. Take the Mathematics Subject Test Level 1 when you have completed three years of college-prep mathematics courses. You may want to time yourself as you take this practice test. It should take you about 6 minutes to complete. On SAT Mathematics Subject Test questions, you receive one point for each correct answer, but you lose a fraction of a point for each incorrect response. Guess only when you can eliminate at least one of the answer choices. 1. In the figure below, m∠1 = m∠2. What is the value of y? 3. What is the contrapositive of the statement “If it is raining, then the football team will win”? Note: Figure not drawn to scale. (A) 10 (C) 40 (E) 60 (B) 30 (D) 50 2. The statement “I will cancel my appointment if and only if I have
a conflict” is true. Which of the following can be concluded? I. If I have a conflict, then I will cancel my appointment. II. If I do not cancel my appointment, then I do not have a conflict. III. If I cancel my appointment, then I have a conflict. (A) I only (C) III only (E) I, II, and III (B) II only (D) I and III (A) If it is not raining, then the football team will not win. (B) If it is raining, then the football team will not win. (C) If the football team wins, then it is raining. (D) If the football team does not win, then it is not raining. (E) If it is not raining, then the football team will win. 4. Given the points D (1, 5) and E (-2, 3), which conclusion is NOT valid? ̶̶ (A) The midpoint of DE is (- 1 _, 4). 2 (B) D and E are collinear. (C) The distance between D and E is √  5. (D) ̶̶ DE ≅ ̶̶ ED (E) D and E are distinct points. 5. For all integers x, what conclusion can be drawn about the value of the expression x 2 __ 2? (A) The value is negative. (B) The value is not negative. (C) The value is even. (D) The value is odd. (E) The value is not a whole number. College Entrance Exam Practice 135 135 �� Gridded Response: Record Your Answer When responding to a gridded-response test item, you must fill out the grid on your answer sheet correctly, or the item will be marked as incorrect. Gridded Response: Solve the equation 1258 - 2 (3x - 72) = -80. The value of x is 247. • Using a pencil, write your answer in the answer boxes at the top of the grid. • Put only one digit in each box. The decimal point has a designated column. • Do not leave a blank box in the middle of an answer. • For each digit, shade the bubble that is in the same column as the digit in the answer box. Gridded Response: The perimeter of a rectangle is 90 in. The width of the rectangle is 18 in
. Find the length of the rectangle in feet. The length of the rectangle is 27 inches, but the problem asks for the measurement in feet. 27 inches = 9 _ 4 • Fractions and mixed numbers cannot be gridded, so you must grid, or 2.25, feet the answer as 2.25. • Using a pencil, write your answer in the answer boxes at the top of the grid. • Put only one digit in each box. The decimal point has a designated column. • Do not leave a blank box in the middle of an answer. • For each digit, shade the bubble that is in the same column as the digit in the answer box. 136 136 Chapter 2 Geometric Reasoning �� Sample C The length of a segment is 897 2 __ gridded this answer as shown. units. A student 5 �� You cannot grid a negative number in a griddedresponse item because the grid does not include the negative sign (-). So if you get a negative answer to a test item, rework the problem. You probably made a math error. Read each statement and answer the questions that follow. Sample A The correct answer to a test item is 1.6. A student gridded this answer as shown. 5. What answer does the grid show? 6. Explain why you cannot grid a fraction or a mixed number. 7. Write the answer 897 2__ 5 be entered in the grid correctly. in a form that could 1. What error did the student make when filling out the grid? 2. Another student got an answer of -1.6. Explain why the student knew this answer was wrong. 8. Another student got an answer of 10,216.5 units. Explain why the student knew this answer was wrong. Sample D The measure of an angle is 48.9°. A student gridded this answer as shown. Sample B The perimeter of a triangle is 2 3 __ gridded this answer as shown. feet. A student 4 9. What answer does the grid show? 10. What error did the student make when filling out the grid? 11. Explain how to correctly grid the answer. 3. What error did the student make when filling out the grid? 4. Explain how to correctly grid the answer. TAKS Tackler 137 137
�� KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1–2 Multiple Choice Use the figure below for Items 1 and 2. In the figure,   DB bisects ∠ADC. 5. A diagonal of a polygon connects nonconsecutive vertices. The table shows the number of diagonals in a polygon with n sides. Number of Sides Number of Diagonals 4 5 6 7 2 5 9 14 If the pattern continues, how many diagonals does a polygon with 8 sides have? 17 19 20 21 6. Which type of transformation maps figure LMNP onto figure L’M’N’P’? 1. Which best describes the intersection of ∠ADB and ∠BDC? Exactly one ray Exactly one point Exactly one angle Exactly one segment 2. Which expression is equal to the measure of ∠ADC? 2 (m∠ADB) 90° - m∠BDC 180° - 2 (m∠ADC) m∠BDC - m∠ADB 3. What is the inverse of the statement, “If a polygon has 8 sides, then it is an octagon”? Reflection Rotation Translation None of these If a polygon is an octagon, then it has 8 sides. If a polygon is not an octagon, then it does not have 8 sides. If an octagon has 8 sides, then it is a polygon. If a polygon does not have 8 sides, then it is not an octagon. 4. Lily conjectures that if a number is divisible by 15, then it is also divisible by 9. Which of the following is a counterexample? 45 50 60 72 7. Miyoko went jogging on July 25, July 28, July 31, and August 3. If this pattern continues, when will Miyoko go jogging next? August 5 August 6 August 7 August 8 8. Congruent segments have equal measures. A segment bisector
divides a segment into ̶̶ two congruent segments. DE at X  XY intersects ̶̶ DE. Which conjecture is valid? and bisects m∠YXD = m∠YXE Y is between D and E. DX = XE DE = YE 138 138 Chapter 2 Geometric Reasoning �� 9. Which statement is true by the Symmetric Property of Congruence? ̶̶ ST ̶̶ ST ≅ 15 + MN = MN + 15 If ∠P ≅ ∠Q, then ∠Q ≅ ∠P. If ∠D ≅ ∠E and ∠E ≅ ∠F, then ∠D ≅ ∠F. �� To find a counterexample for a biconditional statement, write the conditional statement and converse it contains. Then try to find a counterexample for one of these statements. 10. Which is a counterexample for the following biconditional statement? A pair of angles is supplementary if and only if the angles form a linear pair. The measures of supplementary angles add to 180°. A linear pair of angles is supplementary. Complementary angles do not form a linear pair. Two supplementary angles are not adjacent. 11. K is between J and L. The distance between J and K is 3.5 times the distance between K and L. If JK = 14, what is JL? 10.5 18 24.5 49 STANDARDIZED TEST PREP Short Response 16. Solve the equation 2 (AB) + 16 = 24 to find the length of segment AB. Write a justification for each step. 17. Use the given two-column proof to write a flowchart proof. ̶̶ Given: DE ≅ Prove: DE = FG + GH ̶̶ FH Two-column proof: Statements ̶̶ FH ̶̶ DE ≅ 1. Reasons 1. Given 2. DE = FH 2. Def. of ≅ segs. 3. FG + GH = FH 3. Seg. Add. Post. 4. DE = FG + GH 4. Subst. 18. Consider the following conditional statement. If two angles are complementary, then the angles are acute. a. Determine if the conditional is true or false. If false, give a counterexample. b. Write the
converse of the conditional statement. 12. What is the length of the segment connecting the points (-7, -5) and (5, -2)? c. Determine whether the converse is true or false. If false, give a counterexample. √  13 √  53 3 √  17 √  193 Gridded Response 13. A segment has an endpoint at (5, -2). The midpoint of the segment is (2, 2). What is the length of the segment? 14. ∠P measures 30° more than the measure of its supplement. What is the measure of ∠P in degrees? 15. The perimeter of a square field is 1.6 kilometers. What is the area of the field in square kilometers? Extended Response 19. The figure below shows the intersection of two lines. a. Name the linear pairs of angles in the figure. What conclusion can you make about each pair? Explain your reasoning. b. Name the pairs of vertical angles in the figure. What conclusion can you make about each pair? Explain your reasoning. c. Suppose m∠EBD = 90°. What are the measures of the other angles in the figure? Write a two-column proof to support your answer. Cumulative Assessment, Chapters 1–2 139 139 �� T E X A S TAKS Grades 9–11 Obj. 10 �� The Freescale Marathon Every February, runners take to the streets of Austin to participate in a 26-mile marathon. The course travels mostly downhill, with a net drop in elevation of more than 400 feet from beginning to end. But don’t be fooled; many former participants strongly recommend hill training for this course! Choose one or more strategies to solve each problem. 1. During the marathon, a runner maintains a steady pace and completes the first 2.6 miles in 20 minutes. After 1 hour 20 minutes, she has completed 10.4 miles. Make a conjecture about the runner’s average speed in miles per hour. How long do you expect her to take to complete the marathon? 2. The course features fully equipped aid stations with medical support for the complete eight-hour duration of the race. From mile 2 to mile 6, these stations are located every other mile. After that, they are located every mile to the finish. Portable toilets are available at
each mile marker and at the end of the course. At how many points are there both an aid station and portable toilets? �� For 3, use the map. 3. The course includes a straight section along Forty-fifth Street from Shoal Creek Boulevard to Duval Street. The distance from Guadalupe Street to Duval is twice the distance from Burnet Road to Lamar Boulevard. The distance from Lamar to Guadalupe is 210 feet greater than the distance from Shoal Creek to Burnet. What is the distance from Guadalupe to Duval? 140 140 Chapter 2 Geometric Reasoning � � � � � � � � � � � � � � � � � � � � �� Show Caves The region between San Antonio and Austin, known as the Texas Hill Country, is home to six of the seven show caves in the state. A show cave is a cave developed for public use, typically with amenities such as lighting and groomed trails. The seventh show cave in Texas, the Caverns of Sonora, is internationally recognized as one of the most beautiful in the world. Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List Choose one or more strategies to solve each problem. 1. Jared took a tour at least 5 __ 8 mi long and saw a cave that is less than 10,000 ft in length. Which caves might Jared have visited? 2. A travel brochure includes the following statements about Texas show caves. Determine whether each statement is true or false. If false, explain why. a. If you tour a cave that is more Texas Show Caves Cave Length (ft) Approximate Depth (ft) Tour Length (mi) Cascade Caverns Cave Without a Name Caverns of Sonora Inner Space Cavern Longhorn Cavern Natural Bridge Caverns Wonder Cave 1,700 14,211 20,000 15,000 9,850 8,600 1,296 132 89 150 80 23 250 91 0.25 0.
25 1.5 1.2 0.625 0.75 0.08 than 100 ft deep, then you’ll see a cave that is more than 8000 ft in length. b. If you haven’t been to the Caverns of Sonora, then you haven’t seen a cave that is at least 15,000 ft long. c. If you don’t want to walk more than a mile, but you want to see a cave with a depth of at least 150 ft, then you should visit Natural Bridge Caverns. 3. Inner Space Cavern has a second tour, which Ingrid completes in 18 min and 45 s. If Ingrid walks 12,672 ft in 1 h, what is the length of the second tour? Problem Solving on Location 141141 Parallel and Perpendicular Lines 3A Lines with Transversals 3-1 Lines and Angles Lab Explore Parallel Lines and Transversals 3-2 Angles Formed by Parallel Lines and Transversals 3-3 Proving Lines Parallel Lab Construct Parallel Lines 3-4 Perpendicular Lines Lab Construct Perpendicular Lines 3B Coordinate Geometry 3-5 Slopes of Lines Lab Explore Parallel and Perpendicular Lines 3-6 Lines in the Coordinate Plane KEYWORD: MG7 ChProj Sailboats at the Corpus Christi Municipal Marina 142 142 Chapter 3 Vocabulary Match each term on the left with a definition on the right. 1. acute angle A. segments that have the same length 2. congruent angles B. an angle that measures greater than 90° and less than 180° 3. obtuse angle 4. collinear C. points that lie in the same plane D. angles that have the same measure 5. congruent segments E. points that lie on the same line F. an angle that measures greater than 0° and less than 90° Conditional Statements Identify the hypothesis and conclusion of each conditional. 6. If E is on AC, then E lies in plane P. 7. If A is not in plane Q, then A is not on BD. 8. If plane P and plane Q intersect, then they intersect in a line. Name and Classify Angles Name and classify each angle. 9. 10. 11. 12. Angle Relationships Give an example of each angle pair. 13. vertical angles 14. adjacent angles 15.
complementary angles 16. supplementary angles Evaluate Expressions Evaluate each expression for the given value of the variable. 17. 4x + 9 for x = 31 18. 6x - 16 for x = 43 19. 97 - 3x for x = 20 20. 5x + 3x + 12 for x = 17 Solve Multi-Step Equations Solve each equation for x. 21. 4x + 8 = 24 23. 4x + 3x + 6 = 90 22. 2 = 2x - 8 24. 21x + 13 + 14x - 8 = 180 Parallel and Perpendicular Lines 143 143 �� Key Vocabulary/Vocabulario alternate exterior angles alternate interior angles corresponding angles parallel lines ángulos alternos externos ángulos alternos internos ángulos correspondientes líneas paralelas perpendicular bisector mediatriz perpendicular lines líneas perpendiculares same-side interior angles ángulos internos del mismo lado slope transversal pendiente transversal Geometry TEKS G.1.A Geometric Structure* develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, answer the following questions. You may refer to the chapter, the glossary, or a dictionary if you like. 1. The root trans- means “across.” What do you think a transversal of two lines does? 2. The slope of a mountain trail describes the steepness of the climb. What might the slope of a line describe? 3. What does the word corresponding mean? What do you think the term corresponding angles means? 4. What does the word interior mean? What might the phrase “interior of a pair of lines” describe? The word alternate means “to change from one to another.” If two lines are crossed by a third line, where do you think a pair of alternate interior angles might be? 3-2 Tech. Lab Les. 3-1 Les. 3-2 Les. 3-3 ★ 3-3 Geo. Lab Les. 3-4 ★ 3-4 Geo. Lab 3-6 Tech. Lab Les. 3-5 Les. 3-6 G.2.A Geometric Structure* use construction to ★
★ ★ explore attributes of geometric figures and to make conjectures about geometric relationships G.3.C Geometric Structure* use logical reasoning to ★ ★ ★ ★ prove statements are true... G.7.B Dimensionality and the geometry of location* use slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines... G.7.C Dimensionality and the geometry of location* develop and use formulas involving... slope... ★ ★ ★ ★ G.9.A Congruence and the geometry of ★ ★ ★ ★ ★ ★ size* formulate and test conjectures about the properties of parallel and perpendicular lines based on explorations * Knowledge and skills are written out completely on pages TX28–TX35. 144 144 Chapter 3 Study Strategy: Take Effective Notes Taking effective notes is an important study strategy. The Cornell system of note taking is a good way to organize and review main ideas. In the Cornell system, the paper is divided into three main sections. The note-taking column is where you take notes during lecture. The cue column is where you write questions and key phrases as you review your notes. The summary area is where you write a brief summary of the lecture. Step 2: Cues After class, write down key phrases or questions in the left column. Step 3: Summary Use your cues to restate the main points in your own words. 9/4/05 Chapter 2 Lesson 6 page 1 What can you use to just ify steps in a proof? Geometric proof: Start with hypothes is, and then use defs., and thms. to reach posts. conc lus ion. Just ify each step., Step 1: Notes Draw a vertical line about 2.5 inches from the left side of your paper. During class, write your notes about the main points of the lecture in the right column. What k ind of ang les form a l inear pa ir? Linear Pa ir Theorem If 2 ∠ s form a l in. pa ir, then they are supp. Congruent Supp lements Theorem If 2 ∠ s are supp. to the same ∠ (or to 2 ≅ ∠ s), then the 2 ∠ s are ≅. What is true about two supp lements of the same ang le? S u m m a ry : def i n i t i on s, p o stu l ate s, a n d th e o re m s to sh ow th at a c
on c l u s i on i s tr u e. Th e Li n e a r Pa i r Th e o re m s ay s th at two a n g le s th at fo re s u p p le m ent a ry. Th e C on g r u ent S u p p le m ent s Th e o re m s ay s th at two s u p p le m ent s to th e s a m e a n g le a re c on g r u ent. Try This 1. Research and write a paragraph describing the Cornell system of note taking. Describe how you can benefit from using this type of system. 2. In your next class, use the Cornell system of note taking. Compare these notes to your notes from a previous lecture. Parallel and Perpendicular Lines 145 145 3-1 Lines and Angles Objectives Identify parallel, perpendicular, and skew lines. Identify the angles formed by two lines and a transversal. Vocabulary parallel lines perpendicular lines skew lines parallel planes transversal corresponding angles alternate interior angles alternate exterior angles same-side interior angles Who uses this? Card architects use playing cards to build structures that contain parallel and perpendicular planes. Bryan Berg uses cards to build structures like the one at right. In 1992, he broke the Guinness World Record for card structures by building a tower 14 feet 6 inches tall. Since then, he has built structures more than 25 feet tall. Parallel, Perpendicular, and Skew Lines Parallel lines (ǁ) are coplanar and do not intersect. In the figure,   AB ǁ   EF, and   EG ǁ   FH. Perpendicular lines (⊥) intersect at 90° angles. In the figure,   AB ⊥   AE, and   EG ⊥   GH. Skew lines are not coplanar. Skew lines are not parallel and do not intersect. In the figure,   AB and   EG are skew. Parallel planes are planes that do not intersect. In the figure, plane ABE �
� plane CDG. E X A M P L E 1 Identifying Types of Lines and Planes Identify each of the following. Arrows are used to show that   AB ǁ   EF and   EG ǁ   FH. Segments or rays are parallel, perpendicular, or skew if the lines that contain them are parallel, perpendicular, or skew. A a pair of parallel segments ̶̶ KN ǁ ̶̶ PS B a pair of skew segments ̶̶ RS are skew. ̶̶̶ LM and C a pair of perpendicular segments ̶̶̶ MR ⊥ ̶̶ RS D a pair of parallel planes plane KPS ǁ plane LQR Identify each of the following. 1a. a pair of parallel segments 1b. a pair of skew segments 1c. a pair of perpendicular segments 1d. a pair of parallel planes 146 146 Chapter 3 Parallel and Perpendicular Lines �� Angle Pairs Formed by a Transversal TERM EXAMPLE A transversal is a line that intersects two coplanar lines at two different points. The transversal t and the other two lines r and s form eight angles. Corresponding angles lie on the same side of the transversal t, on the same sides of lines r and s. Alternate interior angles are nonadjacent angles that lie on opposite sides of the transversal t, between lines r and s. Alternate exterior angles lie on opposite sides of the transversal t, outside lines r and s. Same-side interior angles or consecutive interior angles lie on the same side of the transversal t, between lines r and s. ∠1 and ∠5 ∠3 and ∠6 ∠1 and ∠8 ∠3 and ∠5 E X A M P L E 2 Classifying Pairs of Angles Give an example of each angle pair. A corresponding angles B alternate interior angles ∠4 and ∠8 ∠4 and ∠6 C alternate exterior angles D same-side interior angles ∠2 and ∠8 ∠4 and ∠5 Give an example of each angle pair. 2a. corresponding angles 2b. alternate interior angles 2c. alternate exterior angles 2d. same-side interior angles E
X A M P L E 3 Identifying Angle Pairs and Transversals Identify the transversal and classify each angle pair. To determine which line is the transversal for a given angle pair, locate the line that connects the vertices. A ∠1 and ∠5 transversal: n; alternate interior angles B ∠3 and ∠6 transversal: m; corresponding angles C ∠1 and ∠4 transversal: ℓ; alternate exterior angles 3. Identify the transversal and classify the angle pair ∠2 and ∠5 in the diagram above. 3- 1 Lines and Angles 147 147 �� THINK AND DISCUSS 1. Compare perpendicular and intersecting lines. 2. Describe the positions of two alternate exterior angles formed by lines m and n with transversal p. 3. GET ORGANIZED Copy the diagram and graphic organizer. In each box, list all the angle pairs of each type in the diagram. 3-1 Exercises Exercises KEYWORD: MG7 3-1 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary? are located on opposite sides of a transversal, between the two ̶̶̶̶ lines that intersect the transversal. (corresponding angles, alternate interior angles, alternate exterior angles, or same-side interior angles Identify each of the following. p. 146 2. one pair of perpendicular segments 3. one pair of skew segments 4. one pair of parallel segments 5. one pair of parallel planes Give an example of each angle pair. p. 147 6. alternate interior angles 7. alternate exterior angles 8. corresponding angles 9. same-side interior angles Identify the transversal and classify each angle pair. p. 147 10. ∠1 and ∠2 11. ∠2 and ∠3 12. ∠2 and ∠4 13. ∠4 and ∠5 148 148 Chapter 3 Parallel and Perpendicular Lines �� Independent Practice For See Exercises Example 14–17 18–21 22–25 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S8 Application Practice p. S30 PRACTICE AND PROBLEM
SOLVING Identify each of the following. 14. one pair of parallel segments 15. one pair of skew segments 16. one pair of perpendicular segments 17. 17. one pair of parallel planes Give an example of each angle pair. 18. same-side interior angles 19. alternate exterior angles 20. corresponding angles 21. alternate interior angles Identify the transversal and classify each angle pair. 22. ∠2 and ∠3 23. ∠4 and ∠5 24. ∠2 and ∠4 25. ∠1 and ∠2 26. Sports A football player runs across the 30-yard line at an angle. He continues in a straight line and crosses the goal line at the same angle. Describe two parallel lines and a transversal in the diagram. Name the type of angle pair shown in each letter. 27. F 28. Z 29. C Entertainment Entertainment In an Ames room, two people of the same height that are standing in different parts of the room appear to be different sizes. Entertainment Entertainment Use the following information for Exercises 30–32. In an Ames room, the floor is tilted and the back wall is closer to the front wall on one side. 30. 30. Name a pair of parallel segments in the diagram. 31. Name a pair of skew segments in the diagram. 32. Name a pair of perpendicular segments in the diagram. 3- 1 Lines and Angles 149 149 �� 33. This problem will prepare you for the Multi-Step TAKS Prep on p 180. Buildings that are tilted like the one shown are sometimes called mystery spots. a. Name a plane parallel to plane KLP, a plane parallel to plane KNP, and a plane parallel to KLM. b. In the diagram, ̶̶ QR is a transversal to ̶̶ PQ and ̶̶ RS. What type of angle pair is ∠PQR and ∠QRS? 34. Critical Thinking Line ℓ is contained in plane P and line m is contained in plane Q. If P and Q are parallel, what are the possible classifications of ℓ and m? Include diagrams to support your answer. Use the diagram for Exercises 35–40. 35. Name a pair of alternate interior angles with transversal n. 36. Name a pair of same-side interior angles with transversal �
��. 37. Name a pair of corresponding angles with transversal m. 38. Identify the transversal and classify the angle pair for ∠3 and ∠7. 39. Identify the transversal and classify the angle pair for ∠5 and ∠8. 40. Identify the transversal and classify the angle pair for ∠1 and ∠6. 41. Aviation Describe the type of lines formed by two planes when flight 1449 is flying from San Francisco to Atlanta at 32,000 feet and flight 2390 is flying from Dallas to Chicago at 28,000 feet. 42. Multi-Step Draw line p, then draw two lines m and n that are both perpendicular to p. Make a conjecture about the relationship between lines m and n. 43. Write About It Discuss a real-world example of skew lines. Include a sketch. 44. Which pair of angles in the diagram are alternate interior angles? ∠1 and ∠5 ∠2 and ∠6 ∠7 and ∠5 ∠2 and ∠3 45. How many pairs of corresponding angles are in the diagram? 2 4 8 16 150 150 Chapter 3 Parallel and Perpendicular Lines ��San FranciscoAtlantaChicagoDallasHolt, Rinehart & WinstonHigh School Mathge07sec03101007a Locator map showing cities3rd proof�� 46. Which type of lines are NOT represented in the diagram? Parallel lines Skew lines Intersecting lines Perpendicular lines 47. For two lines and a transversal, ∠1 and ∠8 are alternate exterior angles, and ∠1 and ∠5 are corresponding angles. Classify the angle pair ∠5 and ∠8. Vertical angles Alternate interior angles Adjacent angles Same-side interior angles 48. Which angles in the diagram are NOT corresponding angles? ∠1 and ∠5 ∠2 and ∠6 ∠4 and ∠8 ∠2 and ∠7 � � � � � � � � CHALLENGE AND EXTEND Name all the angle pairs of each type in the diagram. Identify the transversal for each pair. 49. corresponding 50. alternate interior 51. alternate exterior 52. same-side interior 53. Multi-Step Draw two lines and a transversal such that ∠1 and ∠3 are corresponding angles, ∠1 and ∠2 are alternate interior angles,
and ∠3 and ∠4 are alternate exterior angles. What type of angle pair is ∠2 and ∠4? 54. If the figure shown is folded to form a cube, which faces of the cube will be parallel? � � � � � � � � � � � �� SPIRAL REVIEW Evaluate each function for x = -1, 0, 1, 2, and 3. (Previous course) 55. y = 4 x 2 - 7 56. y = -2 x 2 + 5 57. y = (x + 3) (x - 3) Find the circumference and area of each circle. Use the π key on your calculator and round to the nearest tenth. (Lesson 1-5) 58. �� 59. �� Write a justification for each statement, given that ∠1 and ∠3 are right angles. (Lesson 2-6) 60. ∠1 ≅ ∠3 61. m∠1 + m∠2 = 180° 62. ∠2 ≅ ∠4 � � � � 3- 1 Lines and Angles 151 151 Systems of Equations Algebra Sometimes angle measures are given as algebraic expressions. When you know the relationship between two angles, you can write and solve a system of equations to find angle measures. See Skills Bank page S67 Solving Systems of Equations by Using Elimination Step 1 Write the system so that like terms are under one another. Step 2 Eliminate one of the variables. Step 3 Substitute that value into one of the original equations and solve. Step 4 Write the answers as an ordered pair, (x, y). Step 5 Check your solution. Example 1 Solve for x and y. Since the lines are perpendicular, all of the angles are right angles. To write two equations, you can set each expression equal to 90°. (3x + 2y)° = 90°, (6x - 2y)° = 90° Step 1 Step 2 3x + 2y = 90 6x - 2y = 90 ̶̶̶̶̶̶̶̶ 9x + 0 = 180 Write the system so that like terms are under one another. Add like terms on each side of the equations. The y-term has been eliminated. x = 20 Divide both sides by 9 to solve for x. Step 3 3x + 2y = 90 Write
one of the original equations. 3 (20) + 2y = 90 Substitute 20 for x. 60 + 2y = 90 Simplify. 2y = 30 y = 15 Subtract 60 from both sides. Divide by 2 on both sides. Step 4 (20, 15) Write the solution as an ordered pair. Step 5 Check the solution by substituting 20 for x and 15 for y in the original equations. 3x + 2y = 90 6x - 2y = 90 3(20) + 2(15) 90 6(20) - 2(15) 90 60 + 30 90 120 - 30 90 90 90 ✓ 90 90 ✓ In some cases, before you can do Step 1 you will need to multiply one or both of the equations by a number so that you can eliminate a variable. 152 152 Chapter 3 Parallel and Perpendicular Lines �� Example 2 Solve for x and y. (2x + 4y)° = 72° (5x + 2y)° = 108° Vertical Angles Theorem Linear Pair Theorem The equations cannot be added or subtracted to eliminate a variable. Multiply the second equation by -2 to get opposite y-coefficients. 5x + 2y = 108 → -2 (5x + 2y) = -2 (108) → -10x - 4y = -216 Step 1 Step 2 2x + 4y = 72 -10x - 4y = -216 ̶̶̶̶̶̶̶̶̶̶̶̶ = -144 -8x Write the system so that like terms are under one another. Add like terms on both sides of the equations. The y-term has been eliminated. x = 18 Divide both sides by -8 to solve for x. Step 3 2x + 4y = 72 Write one of the original equations. 2(18) + 4y = 72 Substitute 18 for x. 36 + 4y = 72 Simplify. 4y = 36 y = 9 Subtract 36 from both sides. Divide by 4 on both sides. Step 4 (18, 9) Write the solution as an ordered pair. Step 5 Check the solution by substituting 18 for x and 9 for y in the original equations. 2x + 4y = 72 3(18) + 4(9) 36 + 36 72 72 5x + 2y = 108 5 (18) + 2 (9) 108 90 + 18 108 72 72 ✓ 108 108
✓ Try This TAKS Grades 9–11 Obj. 2, 4, 6 Solve for x and y. 1. 3. 2. 4. On Track for TAKS 153 153 �� 3-2 Use with Lesson 3-2 Activity Explore Parallel Lines and Transversals Geometry software can help you explore angles that are formed when a transversal intersects a pair of parallel lines. TEKS G.9.A Congruence and the geometry of size: formulate and test conjectures about the properties of parallel and perpendicular lines based on explorations.... KEYWORD: MG7 Lab3 1 Construct a line and label two points on the line A and B. 2 Create point C not on AB. Construct a line parallel to   AB through point C. Create another point on this line and label it D. 3 Create two points outside the two parallel lines and label them E and F. Construct transversal   EF. Label the points of intersection G and H. 4 Measure the angles formed by the parallel lines and the transversal. Write the angle measures in a chart like the one below. Drag point E or F and chart with the new angle measures. What relationships do you notice about the angle measures? What conjectures can you make? ∠AGE ∠BGE ∠AGH ∠BGH ∠CHG ∠DHG ∠CHF ∠DHF Angle Measure Measure Try This 1. Identify the pairs of corresponding angles in the diagram. What conjecture can you make about their angle measures? Drag a point in the figure to confirm your conjecture. 2. Repeat steps in the previous problem for alternate interior angles, alternate exterior angles, and same-side interior angles. 3. Try dragging point C to change the distance between the parallel lines. What happens to the angle measures in the figure? Why do you think this happens? 154 154 Chapter 3 Parallel and Perpendicular Lines 3-2 Angles Formed by Parallel Lines and Transversals TEKS G.3.C Geometric structure: use logical reasoning to prove statements are true.... Also G.3.E, G.9.A Objective Prove and use theorems about the angles formed by parallel lines and
a transversal. Who uses this? Piano makers use parallel strings for the higher notes. The longer strings used to produce the lower notes can be viewed as transversals. (See Example 3.) When parallel lines are cut by a transversal, the angle pairs formed are either congruent or supplementary. Postulate 3-2-1 Corresponding Angles Postulate THEOREM HYPOTHESIS CONCLUSION If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. ∠1 ≅ ∠3 ∠2 ≅ ∠4 ∠5 ≅ ∠7 ∠6 ≅ ∠8 E X A M P L E 1 Using the Corresponding Angles Postulate Find each angle measure. A m∠ABC x = 80 m∠ABC = 80° B m∠DEF Corr.  Post. (2x - 45) ° = (x + 30) ° Corr.  Post. x - 45 = 30 x = 75 m∠DEF = x + 30 Subtract x from both sides. Add 45 to both sides. = 75 + 30 = 105° Substitute 75 for x. 1. Find m∠QRS. Remember that postulates are statements that are accepted without proof. Since the Corresponding Angles Postulate is given as a postulate, it can be used to prove the next three theorems. 3- 2 Angles Formed by Parallel Lines and Transversals 155 155 �� Theorems Parallel Lines and Angle Pairs THEOREM HYPOTHESIS CONCLUSION If a transversal is perpendicular to two parallel lines, all eight angles are congruent. 3-2-2 Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. 3-2-3 Alternate Exterior Angles Theorem If two parallel lines are cut by a transversal, then the two pairs of alternate exterior angles are congruent. 3-2-4 Same-Side Interior Angles Theorem If two parallel lines are cut by a transversal, then the two pairs of same-side interior angles are supplementary. ∠1 ≅ ∠3 ∠2 ≅ ∠4 ∠5 ≅ ∠7 ∠
6 ≅ ∠8 m∠1 + m∠4 = 180° m∠2 + m∠3 = 180° You will prove Theorems 3-2-3 and 3-2-4 in Exercises 25 and 26. PROOF PROOF Alternate Interior Angles Theorem Given: ℓ ǁ m Prove: ∠ 2 ≅ ∠3 Proof: E X A M P L E 2 Finding Angle Measures Find each angle measure. A m∠EDF x = 125 m∠EDF = 125° Alt. Ext.  Thm. B m∠TUS 13x° + 23x° = 180° Same-Side Int.  Thm. 36x = 180 x = 5 m∠TUS = 23 (5) = 115° Combine like terms. Divide both sides by 36. Substitute 5 for x. 2. Find m∠ABD. 156 156 Chapter 3 Parallel and Perpendicular Lines �� Parallel Lines and Transversals When I solve problems with parallel lines and transversals, I remind myself that every pair of angles is either congruent or supplementary. If r ǁ s, all the acute angles are congruent and all the obtuse angles are congruent. The acute angles are supplementary to the obtuse angles. Nancy Martin East Branch High School E X A M P L E 3 Music Application The treble strings of a grand piano are parallel. Viewed from above, the bass strings form transversals to the treble strings. Find x and y in the diagram. By the Alternate Exterior Angles Theorem, (25x + 5y) ° = 125°. By the Corresponding Angles Postulate, (25x + 4y) ° = 120°. 25x + 5y = 125 - (25x + 4y = 120) ̶̶̶̶̶̶̶̶̶̶̶̶ y = 5 25x + 5 (5) = 125 x = 4, y = 5 Subtract the second equation from the first equation. Substitute 5 for y in 25x + 5y = 125. Simplify and solve for x. 3. Find the measures of the acute angles in the diagram.
THINK AND DISCUSS 1. Explain why a transversal that is perpendicular to two parallel lines forms eight congruent angles. 2. GET ORGANIZED Copy the diagram and graphic organizer. Complete the graphic organizer by explaining why each of the three theorems is true. 3- 2 Angles Formed by Parallel Lines and Transversals 157 157 �� 3-2 Exercises Exercises GUIDED PRACTICE Find each angle measure. p. 155 1. m∠JKL 2. m∠BEF KEYWORD: MG7 3-2 KEYWORD: MG7 Parent. m∠1 4. m∠CBY p. 156. Safety The railing of p. 157 a wheelchair ramp is parallel to the ramp. Find x and y in the diagram. Independent Practice For See Exercises Example 6–7 8–11 12 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S8 Application Practice p. S30 PRACTICE AND PROBLEM SOLVING Find each angle measure. 6. m∠KLM 7. m∠VYX 8. m∠ ABC 9. m∠EFG 10. m∠PQR 11. m∠STU 158 158 Chapter 3 Parallel and Perpendicular Lines �� 12. Parking In the parking lot shown, the lines that mark the width of each space are parallel. m∠1 = (2x - 3y) ° m∠2 = (x + 3y) ° Find x and y. Find each angle measure. Justify each answer with a postulate or theorem. 13. m∠1 14. m∠2 15. m∠3 16. m∠4 17. m∠5 18. m∠6 19. m∠7 Algebra State the theorem or postulate that is related to the measures of the angles in each pair. Then find the angle measures. 20. m∠1 = (7x + 15) °, m
∠2 = (10x - 9) ° 21. m∠3 = (23x + 11) °, m∠4 = (14x + 21) ° 22. m∠4 = (37x - 15) °, m∠5 = (44x - 29) ° 23. m∠1 = (6x + 24) °, m∠4 = (17x - 9) ° 24. Architecture The Luxor Hotel in Las Vegas, Nevada, is a 30-story pyramid. The hotel uses an elevator called an inclinator to take people up the side of the pyramid. The inclinator travels at a 39° angle. Which theorem or postulate best illustrates the angles formed by the path of the inclinator and each parallel floor? (Hint: Draw a picture.) 25. Complete the two-column proof of the Alternate Exterior Angles Theorem. Given: ℓ ǁ m Prove: ∠1 ≅ ∠2 Proof: Statements Reasons Architecture Architecture The Luxor hotel is 600 feet wide, 600 feet long, and 350 feet high. The atrium in the hotel measures 29 million cubic feet. 1. ℓ ǁ m 2. a.? ̶̶̶̶̶ 3. ∠3 ≅ ∠2 4. c.? ̶̶̶̶̶ 1. Given 2. Vert.  Thm. 3. b. 4. d.? ̶̶̶̶̶? ̶̶̶̶̶ 26. Write a paragraph proof of the Same-Side Interior Angles Theorem. Given: r ǁ s Prove: m∠1 + m∠2 = 180° Draw the given situation or tell why it is impossible. 27. Two parallel lines are intersected by a transversal so that the corresponding angles are supplementary. 28. Two parallel lines are intersected by a transversal so that the same-side interior angles are complementary. 3- 2 Angles Formed by Parallel Lines and Transversals 159 159 �� 29. This problem will prepare you for the Multi-Step TAKS Prep on page 180. In the diagram, which represents the side view of a mystery spot, m∠SRT = 25°.   RT is a transversal to  
PS and   QR. a. What type of angle pair is ∠QRT and ∠STR? b. Find m∠STR. Use a theorem or postulate to justify your answer. 30. Land Development A piece of property lies between two parallel streets as shown. m∠1 = (2x + 6) °, and m∠2 = (3x + 9) °. What is the relationship between the angles? What are their measures? 31. /////ERROR ANALYSIS///// In the figure, m∠ABC = (15x + 5) °, and m∠BCD = (10x + 25) °. Which value of m∠BCD is incorrect? Explain. 32. Critical Thinking In the diagram, ℓ ǁ m. Explain why x _ y = 1. 33. Write About It Suppose that lines ℓ and m are intersected by transversal p. One of the angles formed by ℓ and p is congruent to every angle formed by m and p. Draw a diagram showing lines ℓ, m, and p, mark any congruent angles that are formed, and explain what you know is true. 34. m∠RST = (x + 50) °, and m∠STU = (3x + 20) °. Find m∠RVT. 15° 27.5° 65° 77.5° 160 160 Chapter 3 Parallel and Perpendicular Lines �� 35. For two parallel lines and a transversal, m∠1 = 83°. For which pair of angle measures is the sum the least? ∠1 and a corresponding angle ∠1 and a same-side interior angle ∠1 and its supplement ∠1 and its complement 36. Short Response Given a ǁ b with transversal t, explain why �
�1 and ∠3 are supplementary. � � � � � � CHALLENGE AND EXTEND Multi-Step Find m∠1 in each diagram. (Hint: Draw a line parallel to the given parallel lines.) 37. �� 38. � �� 39. Find x and y in the diagram. Justify your answer. � 40. Two lines are parallel. The measures of two corresponding angles are a° and 2b°, and the measures of two same-side interior angles are a° and b°. Find the value of a. � �� SPIRAL REVIEW If the first quantity increases, tell whether the second quantity is likely to increase, decrease, or stay the same. (Previous course) 41. time in years and average cost of a new car 42. age of a student and length of time needed to read 500 words Use the Law of Syllogism to draw a conclusion from the given information. (Lesson 2-3) 43. If two angles form a linear pair, then they are supplementary. If two angles are supplementary, then their measures add to 180°. ∠1 and ∠2 form a linear pair. 44. If a figure is a square, then it is a rectangle. If a figure is a rectangle, then its sides are perpendicular. Figure ABCD is a square. Give an example of each angle pair. (Lesson 3-1) 45. alternate interior angles 46. alternate exterior angles 47. same-side interior angles � � � � � � � � 3- 2 Angles Formed by Parallel Lines and Transversals 161 161 3-3 Proving Lines Parallel TEKS G.3.C Geometric structure: use logical reasoning to prove statements are true.... Also G.1.A, G.3.E, G.9.A Objective Use the angles formed by a transversal to prove two lines are parallel. Who uses this? Rowers have to keep the oars on each side parallel in order to travel in a straight line. (See Example 4.) Recall that the converse of a theorem is found by exchanging the hypothesis and conclusion. The converse of a theorem is not automatically true. If it is true, it must be stated as a postulate or proved as a separate theorem. Postulate 3-3-1 Converse of the Corresponding Angles Postulate THEOREM HYPOTHESIS
CONCLUSION If two coplanar lines are cut by a transversal so that a pair of corresponding angles are congruent, then the two lines are parallel. ∠1 ≅ ∠ Using the Converse of the Corresponding Angles Postulate Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ ǁ m. A ∠1 ≅ ∠5 ∠1 ≅ ∠5 ℓ ǁ m ∠1 and ∠5 are corresponding angles. Conv. of Corr. ∠s Post. B m∠4 = (2x + 10) °, m∠8 = (3x - 55) °, x = 65 m∠4 = 2 (65) + 10 = 140 m∠8 = 3 (65) - 55 = 140 m∠4 = m∠8 ∠4 ≅ ∠8 ℓ ǁ m Substitute 65 for x. Substitute 65 for x. Trans. Prop. of Equality Def. of ≅  Conv. of Corr.  Post. Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ ǁ m. 1a. m∠1 = m∠3 1b. m∠7 = (4x + 25) °, m∠5 = (5x + 12) °, x = 13 162 162 Chapter 3 Parallel and Perpendicular Lines �� Postulate 3-3-2 Parallel Postulate Through a point P not on line ℓ, there is exactly one line parallel to ℓ. The Converse of the Corresponding Angles Postulate is used to construct parallel lines. The Parallel Postulate guarantees that for any line ℓ, you can always construct a parallel line through a point that is not on ℓ. Construction Parallel Lines  Draw a line ℓ and a point P that is not on ℓ.  Draw a line m through P that intersects ℓ. Label the angle 1.  Construct an angle congruent to ∠1 at P. By the converse of the Corresponding Angles Postulate, ℓ ǁ n. Theorems Proving Lines Parallel THEOREM HYPOTHESIS CONCLUSION
3-3-3 Converse of the Alternate ∠1 ≅ ∠2 Interior Angles Theorem If two coplanar lines are cut by a transversal so that a pair of alternate interior angles are congruent, then the two lines are parallel. 3-3-4 Converse of the Alternate ∠3 ≅ ∠4 Exterior Angles Theorem If two coplanar lines are cut by a transversal so that a pair of alternate exterior angles are congruent, then the two lines are parallel. 3-3-5 Converse of the Same-Side m∠5 + m∠6 = 180° Interior Angles Theorem If two coplanar lines are cut by a transversal so that a pair of same-side interior angles are supplementary, then the two lines are parallel You will prove Theorems 3-3-3 and 3-3-5 in Exercises 38–39. 3- 3 Proving Lines Parallel 163 163 �� PROOF PROOF Converse of the Alternate Exterior Angles Theorem Given: ∠1 ≅ ∠2 Prove: ℓ ǁ m Proof: It is given that ∠1 ≅ ∠2. Vertical angles are congruent, so ∠1 ≅ ∠3. By the Transitive Property of Congruence, ∠2 ≅ ∠3. So ℓ ǁ m by the Converse of the Corresponding Angles Postulate. E X A M P L E 2 Determining Whether Lines are Parallel Use the given information and the theorems you have learned to show that r ǁ s. A ∠2 ≅ ∠6 ∠2 ≅ ∠6 r ǁ s ∠2 and ∠6 are alternate interior angles. Conv. of Alt. Int.  Thm. B m∠6 = (6x + 18) °, m∠7 = (9x + 12) °, x = 10 m∠6 = 6x + 18 = 6 (10) + 18 = 78° Substitute 10 for x. m∠7 = 9x + 12 = 9 (10) + 12 = 102° m∠6 + m∠7 = 78° + 102° Substitute 10 for x. = 180° ∠6 and ∠7 are
same-side interior angles. r ǁ s Conv. of Same-Side Int.  Thm. Refer to the diagram above. Use the given information and the theorems you have learned to show that r ǁ s. 2a. m∠4 = m∠8 2b. m∠3 = 2x°, m∠7 = (x + 50) °, x = 50 E X A M P L E 3 Proving Lines Parallel Given: ℓ ǁ m, ∠1 ≅ ∠3 Prove: r ǁ p Proof: Statements Reasons 1. ℓ ǁ m 2. ∠1 ≅ ∠2 3. ∠1 ≅ ∠3 4. ∠2 ≅ ∠3 5. r ǁ p 1. Given 2. Corr.  Post. 3. Given 4. Trans. Prop. of ≅ 5. Conv. of Alt. Ext.  Thm. 3. Given: ∠1 ≅ ∠4, ∠3 and ∠4 are supplementary. Prove: ℓ ǁ m 164 164 Chapter 3 Parallel and Perpendicular Lines �� E X A M P L E 4 Sports Application During a race, all members of a rowing team should keep the oars parallel on each side. If m∠1 = (3x + 13) °, m∠2 = (5x - 5) °, and x = 9, show that the oars are parallel. A line through the center of the boat forms a transversal to the two oars on each side of the boat. ∠1 and ∠2 are corresponding angles. If ∠1 ≅ ∠2, then the oars are parallel. Substitute 9 for x in each expression: m∠1 = 3x + 13 = 3 (9) + 13 = 40° Substitute 9 for x in each expression. m∠2 = 5x - 5 = 5 (9) - 5 = 40° m∠1 = m∠2, so ∠1 ≅ ∠2. The corresponding angles are congruent, so the oars are parallel by the Converse of the Corresponding Angles Postulate. 4. What if…? Suppose the corresponding angles on the opposite side of
the boat measure (4y - 2) ° and (3y + 6) °, where y = 8. Show that the oars are parallel. THINK AND DISCUSS 1. Explain three ways of proving that two lines are parallel. 2. If you know m∠1, how could you use the measures of ∠5, ∠6, ∠7, or ∠8 to prove m ǁ n? 3. GET ORGANIZED Copy and complete the graphic organizer. Use it to compare the Corresponding Angles Postulate with the Converse of the Corresponding Angles Postulate. 3- 3 Proving Lines Parallel 165 165 21�� KEYWORD: MG7 3-3 KEYWORD: MG7 Parent 3-3 Exercises Exercises GUIDED PRACTICE. 162 Use the Converse of the Corresponding Angles Postulate and the given information to show that p ǁ q. 1. ∠4 ≅ ∠5 2. m∠1 = (4x + 16) °, m∠8 = (5x - 12) °, x = 28 3. m∠4 = (6x - 19) °, m∠5 = (3x + 14) °, x = 11 Use the theorems and given information to show that r ǁ s. p. 164 4. ∠1 ≅ ∠5 5. m∠3 + m∠4= 180° 6. ∠3 ≅ ∠7 7. m∠4 = (13x - 4) °, m∠8 = (9x + 16) °, x = 5 8. m∠8 = (17x + 37) °, m∠7 = (9x - 13) °, x = 6 9. m∠2 = (25x + 7) °, m∠6 = (24x + 12) °, 10. Complete the following two-column proof. p. 164 Given: ∠1 ≅ ∠2, ∠3 ≅ ∠1 Prove: XY ǁ WV Proof: Statements Reasons 1. ∠1 ≅ ∠2, ∠3 ≅ ∠1 1. Given 2. ∠2 ≅
∠3 3. b.? ̶̶̶̶̶ 2. a. 3. c.? ̶̶̶̶̶? ̶̶̶̶̶ 11. Architecture In the fire escape, p. 165 m∠1 = (17x + 9) °, m∠2 = (14x + 18) °, and x = 3. Show that the two landings are parallel. PRACTICE AND PROBLEM SOLVING Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ ǁ m. 12. ∠3 ≅ 7 13. m∠4 = 54°, m∠8 = (7x + 5) °, x = 7 14. m∠2 = (8x + 4) °, m∠6 = (11x - 41) °, x = 15 15. m∠1 = (3x + 19) °, m∠5 = (4x + 7) °, x = 12 166 166 Chapter 3 Parallel and Perpendicular Lines �� Independent Practice Use the theorems and given information to show that n ǁ p. For See Exercises Example 12–15 16–21 22 23 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S8 Application Practice p. S30 16. ∠3 ≅ ∠6 17. ∠2 ≅ ∠7 18. m∠4 + m∠6 = 180° 19. m∠1 = (8x - 7) °, m∠8 = (6x + 21) °, x = 14 20. m∠4 = (4x + 3) °, m∠5 = (5x -22) °, x = 25 21. m∠3 = (2x + 15) °, m∠5 = (3x + 15) °, x = 30 22. Complete the following two-column proof. Given: Prove: ̶̶ AB ǁ ̶̶ BC ǁ ̶̶ CD, ∠1 ≅ ∠2, ∠3 ≅ ∠4 ̶̶ DE Proof: Statements Reasons ̶̶ AB ǁ ̶̶ CD 1. 2. ∠1 ≅ ∠
3 3. ∠1 ≅ ∠2, ∠3 ≅ ∠4 4. ∠2 ≅ ∠4 5. d.? ̶̶̶̶̶̶ 1. Given 2. a. 3. b. 4. c. 5. e.? ̶̶̶̶̶̶? ̶̶̶̶̶̶? ̶̶̶̶̶̶? ̶̶̶̶̶̶ 23. Art Edmund Dulac used perspective when drawing the floor titles in this illustration for The Wind’s Tale by Hans Christian Andersen. Show that DJ ǁ EK if m∠1 = (3x + 2) °, m∠2 = (5x - 10) °, and x = 6. � � � � � � � � � � � � Name the postulate or theorem that proves that ℓ ǁ m. 24. ∠8 ≅ ∠6 25. ∠8 ≅ ∠4 26. ∠2 ≅ ∠6 27. ∠7 ≅ ∠5 28. ∠3 ≅ ∠7 29. m∠2 + m∠3 = 180° For the given information, tell which pair of lines must be parallel. Name the postulate or theorem that supports your answer. 30. m∠2 = m∠10 31. m∠8 + m∠9 = 180° 32. ∠1 ≅ ∠7 33. m∠10 = m∠6 34. ∠11 ≅ ∠5 35. m∠2 + m∠5 = 180° 36. Multi-Step Two lines are intersected by a transversal so that ∠1 and ∠2 are corresponding angles, ∠1 and ∠3 are alternate exterior angles, and ∠3 and ∠4 are corresponding angles. If ∠2 ≅ ∠4, what theorem or postulate can be used to prove the lines parallel? 3- 3 Proving Lines Parallel 167 167 �� 37. This problem will prepare you for the Multi-Step TAKS Prep on page 180. In the diagram, which represents the side view of a mystery spot, m∠SRT = 25°, and m∠SUR = 65°. a. Name a same-side interior angle of ∠S
UR ̶̶ for lines   SU and   RT with transversal RU. What is its measure? Explain your reasoning. b. Prove that   SU and   RT are parallel. 38. Complete the flowchart proof of the Converse of the Alternate Interior Angles Theorem. Given: ∠2 ≅ ∠3 Prove: ℓ ǁ m Proof: 39. Use the diagram to write a paragraph proof of the Converse of the Same-Side Interior Angles Theorem. Given: ∠1 and ∠2 are supplementary. Prove: ℓ ǁ m 40. Carpentry A plumb bob is a weight hung at the end of a string, called a plumb line. The weight pulls the string down so that the plumb line is perfectly vertical. Suppose that the angle formed by the wall and the roof is 123° and the angle formed by the plumb line and the roof is 123°. How does this show that the wall is perfectly vertical? 41. Critical Thinking Are the Reflexive, Symmetric, and Transitive Properties true for parallel lines? Explain why or why not. Reflexive: ℓ ǁ ℓ Symmetric: If ℓ ǁ m, then m ǁ ℓ. Transitive: If ℓ ǁ m and m ǁ n, then ℓ ǁ n. 42. Write About It Does the information given in the diagram allow you to conclude that a ǁ b? Explain. 43. Which postulate or theorem can be used to prove ℓ ǁ m? Converse of the Corresponding Angles Postulate Converse of the Alternate Interior Angles Theorem Converse of the Alternate Exterior Angles Theorem Converse of the Same-Side Interior Angles Theorem 168 168 Chapter 3 Parallel and Perpendicular Lines ��PlumblineWallRoof123˚123˚�� 44. Two coplanar lines are cut by a transversal. Which condition does NOT guarantee that the two lines are parallel? A pair of alternate interior angles are congruent. A pair of same-side interior
angles are supplementary. A pair of corresponding angles are congruent. A pair of alternate exterior angles are complementary. 45. Gridded Response Find the value of x so that ℓ ǁ m. �� CHALLENGE AND EXTEND Determine which lines, if any, can be proven parallel using the given information. Justify your answers. 46. ∠1 ≅ ∠15 48. ∠3 ≅ ∠7 50. ∠6 ≅ ∠8 47. ∠8 ≅ ∠14 49. ∠8 ≅ ∠10 51. ∠13 ≅ ∠11 52. m∠12 + m∠15 = 180° 53. m∠5 + m∠8 = 180° 54. Write a paragraph ̶̶ BD. proof that ̶̶ AE ǁ � � � �� Use the diagram for Exercises 55 and 56. 55. Given: m∠2 + m∠3 = 180° Prove: ℓ ǁ m 56. Given: m∠2 + m∠5 = 180° Prove: ℓ ǁ n � � � � � � � � � � � � � � � � �� SPIRAL REVIEW Solve each equation for the indicated variable. (Previous course) 57. a - b = -c, for a 58. y = 1 _ 2 x - 10, for x 59. 4y + 6x = 12, for y Write the converse, inverse, and contrapositive of each conditional statement. Find the truth value of each. (Lesson 2-2) 60. If an animal is a bat, then it has wings. 61. If a polygon is a triangle, then it has exactly three sides. 62. If the digit in the ones place of a whole number is 2, then the number is even. Identify each of the following. (Lesson 3-1) � 63. one pair of parallel segments 64. one pair of skew segments 65. one pair of perpendicular segments � � � � 3- 3 Proving Lines Parallel 169 169 3-3 Construct Parallel Lines In Lesson 3-3, you learned one method of constructing parallel lines using a compass and straightedge. Another method, called
the rhombus method, uses a property of a figure called a rhombus, which you will study in Chapter 6. The rhombus method is shown below. Use with Lesson 3-3 TEKS G.2.A Geometric structure: use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. Also G.9.A Activity 1 1 Draw a line ℓ and a point P not on the line. 2 Choose a point Q on the line. Place your compass point at Q and draw an arc through P that intersects ℓ. Label the intersection R. 3 Using the same compass setting as the 4 Draw PS ǁ ℓ. first arc, draw two more arcs: one from P, the other from R. Label the intersection of the two arcs S. Try This 1. Repeat Activity 1 using a different point not on the line. Are your results the same? 2. Using the lines you constructed in Problem 1, draw transversal PQ. Verify that the lines are parallel by using a protractor to measure alternate interior angles. 3. What postulate ensures that this construction is always possible? 4. A rhombus is a quadrilateral with four congruent sides. Explain why this method is called the rhombus method. 170 170 Chapter 3 Parallel and Perpendicular Lines �� Activity 2 1 Draw a line ℓ and point P on a piece of 2 Fold the paper through P so that both sides patty paper. of line ℓ match up 3 Crease the paper to form line m. P should 4 Fold the paper again through P so that be on line m. both sides of line m match up. 5 Crease the paper to form line n. Line n is parallel to line ℓ through P. Try This 5. Repeat Activity 2 using a point in a different place not on the line. Are your results the same? 6. Use a protractor to measure corresponding angles. How can you tell that the lines are parallel? 7. Draw a triangle and construct a line parallel to one side through the vertex that is not on that side. 8. Line m is perpendicular to both ℓ and n. Use this statement to complete the following conjecture: If two lines in a plane are perpendicular to the same line, then?. ̶̶̶̶̶̶
̶̶̶̶ 3- 3 Geometry Lab 171 171 3-4 Perpendicular Lines TEKS G.1.A Geometric structure:... connecting definitions... logical reasoning, and theorems. Also G.2.A, G.3.C, G.3.E, G.9.A Objective Prove and apply theorems about perpendicular lines. Vocabulary perpendicular bisector distance from a point to a line Why learn this? Rip currents are strong currents that flow away from the shoreline and are perpendicular to it. A swimmer who gets caught in a rip current can get swept far out to sea. (See Example 3.) The perpendicular bisector of a segment is a line perpendicular to a segment at the segment’s midpoint. A construction of a perpendicular bisector is shown below. Construction Perpendicular Bisector of a Segment  ̶̶ AB. Open the compass Draw wider than half of AB and draw an arc centered at A.  Using the same compass setting, draw an arc centered at B that intersects the first arc at C and D.  Draw   CD.   CD is the perpendicular bisector of ̶̶ AB. The shortest segment from a point to a line is perpendicular to the line. This fact is used to define the distance from a point to a line as the length of the perpendicular segment from the point to the line. E X A M P L E 1 Distance From a Point to a Line A Name the shortest segment from P to    AC. The shortest distance from a point to a line is the length of the perpendicular segment, ̶̶ PB is the shortest segment from P to   AC. so B Write and solve an inequality for x. ̶̶ PB is the shortest segment. Substitute x + 3 for PA and 5 for PB. Subtract 3 from both sides of the inequality. PA > PB x + 3 > 5 - 3 - 3 ̶̶̶ ̶̶̶̶ x > 2 1a. Name the shortest segment from A to   BC. 1b. Write and solve an inequality for x. 172 172 Chapter 3 Parallel and Perpendicular Lines �� Theorems THEOREM DIAGRAM EXAMPLE 3
-4-1 If two intersecting lines form a linear pair of congruent angles, then the lines are perpendicular. (2 intersecting lines form lin. pair of ≅  → lines ⊥.) 3-4-2 Perpendicular Transversal Theorem In a plane, if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other line. If two coplanar lines are perpendicular to the same line, then the two lines are parallel to each other. (2 lines ⊥ to same line → 2 lines ǁ.) 3-4- You will prove Theorems 3-4-1 and 3-4-3 in Exercises 37 and 38. PROOF PROOF Perpendicular Transversal Theorem Given: BC ǁ DE, AB ⊥ BC Prove: AB ⊥ DE Proof: It is given that BC ǁ DE, so ∠ABC ≅ ∠BDE by the Corresponding Angles Postulate. It is also given that   AB ⊥   BC, so m∠ABC = 90°. By the definition of congruent angles, m∠ABC = m∠BDE, so m∠BDE = 90° by the Transitive Property of Equality. By the definition of perpendicular lines,   AB ⊥   DE. E X A M P L E 2 Proving Properties of Lines Write a two-column proof. Given:   AD ǁ   BC,   AD ⊥   AB,   BC ⊥   DC Prove:   AB ǁ  �
�� DC Proof: Statements Reasons 1.   AD ǁ   BC,   BC ⊥   DC 1. Given 2.   AD ⊥   DC 3.   AD ⊥   AB 4.   AB ǁ   DC 2. ⊥ Transv. Thm. 3. Given 4. 2 lines ⊥ to same line → 2 lines ǁ. 2. Write a two-column proof. Given: ∠EHF ≅ ∠HFG,   FG ⊥   GH Prove:   EH ⊥   GH 3- 4 Perpendicular Lines 173 173 �� E X A M P L E 3 Oceanography Application Oceanography The National Weather Service in Brownsville provides a rip current forecast for South Padre Island and other Texas beaches. Park officials also post flags on the beach to warn of rip current danger. Source: www.ripcurrents. noaa.gov Rip currents may be caused by a sandbar parallel to the shoreline. Waves cause a buildup of water between the sandbar and the shoreline. When this water breaks through the sandbar, it flows out in a direction perpendicular to the sandbar. Why must the rip current be perpendicular to the shoreline? The rip current forms a transversal to the shoreline and the sandbar. The shoreline and the sandbar are parallel, and the rip current is perpendicular to the sandbar. So by the Perpendicular Transversal Theorem, the rip current is perpendicular to the shoreline. 3. A swimmer who gets caught in a rip current should swim in a direction perpendicular to the current. Why should the path of the swimmer be parallel to the shoreline? THINK AND DISCUSS 1. Describe what happens if two intersecting lines form a linear pair of congruent angles. 2. Explain why a transversal
that is perpendicular to two parallel lines forms eight congruent angles. 3. GET ORGANIZED Copy and complete the graphic organizer. Use the diagram and the theorems from this lesson to complete the table. 174 174 Chapter 3 Parallel and Perpendicular Lines Rip currentSandbarSandbarShoreline�� 3-4 Exercises Exercises KEYWORD: MG7 3-4 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary   CD is the perpendicular bisector of ̶̶ AB.   CD intersects What can you say about ̶̶ AB and   CD? What can you say about ̶̶ AC and ̶̶ AB at C. ̶̶ BC?. Name the shortest segment from p. 172 point E to   AD. 3. Write and solve an inequality for x. Complete the two-column proof. p. 173 Given: ∠ABC ≅ ∠CBE,   DE ⊥   AF Prove:   CB ǁ   DE Proof: Statements Reasons 1. ∠ABC ≅ ∠CBE 1. Given 2.   CB ⊥   AF 3. b.? ̶̶̶̶̶̶ 4.   CB ǁ   DE 2. a.? ̶̶̶̶̶̶ 3. Given 4. c.? ̶̶̶̶̶. 174 5. Sports The center line in a tennis court is perpendicular to both service lines. Explain why the service lines must be parallel to each other. Independent Practice For See Exercises Example 6–7 8 9 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S8 Application Practice p. S30 PRACTICE AND PROBLEM SOLVING 6. Name the shortest segment from point W to ̶̶ XZ. 7. Write and
solve an inequality for x. 8. Complete the two-column proof below. Given:   AB ⊥   BC, m∠1 + m∠2 = 180° Prove:   BC ⊥   CD Proof: Statements Reasons 1.   AB ⊥   BC 2. m∠1 + m∠2 = 180° 1. Given 2. a.? ̶̶̶̶̶ 3. ∠1 and ∠2 are supplementary. 3. Def. of supplementary 4. b.? ̶̶̶̶̶ 5.   BC ⊥   CD 4. Converse of the Same-Side Interior Angles Theorem 5. c.? ̶̶̶̶̶ 3- 4 Perpendicular Lines 175 175 ��ServicelineCenterlineServicelinege07se_c03l04003ad2ndpass3/18/05NPatel�� 9. Music The frets on a guitar are all perpendicular to one of the strings. Explain why the frets must be parallel to each other. For each diagram, write and solve an inequality for x. 10. 11. Multi-Step Solve to find x and y in each diagram. 12. 14. 13. �� 15. Determine if there is enough information given in the diagram to prove each statement. 16. ∠1 ≅ ∠2 17. ∠1 ≅ ∠3 18. ∠2 ≅ ∠3 19. ∠2 ≅ ∠4 20. ∠3 ≅ ∠4 21. ∠3 ≅ ∠5 22. Critical Thinking Are the Reflexive, Symmetric, and Transitive Properties true for perpendicular lines? Explain why or why not. Reflexive: ℓ ⊥ ℓ Symmetric: If ℓ ⊥ m, then m ⊥ ℓ. Transitive: If ℓ ⊥ m and m ⊥ n, then ℓ ⊥ n. 23
. This problem will prepare you for the Multi-Step TAKS Prep on page 180. a mystery spot, In the diagram, which represents the side view of ̶̶ PQ, ̶̶ PS ⊥ ̶̶ PQ ǁ ̶̶ RS. ̶̶ RS, and a. Prove ̶̶ PS ǁ ̶̶ QR. ̶̶ QR ⊥ ̶̶ RS and ̶̶ PS. ̶̶ QR ⊥ ̶̶ PQ ⊥ b. Prove 176 176 Chapter 3 Parallel and Perpendicular Lines �� 24. Geography Felton Avenue, Arlee Avenue, and Viehl Avenue are all parallel. Broadway Street is perpendicular to Felton Avenue. Use the satellite photo and the given information to determine the values of x and y. 25. Estimation Copy the diagram onto a grid with 1 cm by 1 cm squares. Estimate the distance from point P to line ℓ. �� 26. Critical Thinking Draw a figure to show that Theorem 3-4-3 is not true if the lines are not in the same plane. 27. Draw a figure in which perpendicular bisector of ̶̶ AB. ̶̶ AB is a perpendicular bisector of ̶̶ XY but ̶̶ XY is not a 28. Write About It A ladder is formed by rungs that are perpendicular to the sides of the ladder. Explain why the rungs of the ladder are parallel. �� Construction Construct a segment congruent to each given segment and then �� construct its perpendicular bisector. 29. 30. 31. Which inequality is correct for the given diagram? 2x + 5 < 3x x > 1 2x + 5 > 3x x > 5 32. In the diagram, ℓ ⊥ m. Find x and y. x = 5, y = 7 x = 7, y = 5 x = 90, y = 90 x = 10, y = 5 33. If ℓ ⊥ m, which statement is NOT correct? m∠2 = 90° m∠1 + m∠2 = 180° ∠1 ≅ ∠2 ∠1 �
�� ∠2 3- 4 Perpendicular Lines 177 177 �� 34. In a plane, both lines m and n are perpendicular to both lines p and q. Which conclusion CANNOT be made All angles formed by lines m, n, p, and q are congruent. 35. Extended Response Lines m and n are parallel. Line p intersects line m at A and line n at B, and is perpendicular to line m. a. What is the relationship between line n and line p? Draw a diagram to support your answer. b. What is the distance from point A to line n? What is the distance from point B to line m? Explain. c. How would you define the distance between two parallel lines in a plane? CHALLENGE AND EXTEND 36. Multi-Step Find m∠1 in the diagram. (Hint: Draw a line parallel to the given parallel lines.) 37. Prove Theorem 3-4-1: If two intersecting lines form a linear pair of congruent angles, then the two lines are perpendicular. 38. Prove Theorem 3-4-3: If two coplanar lines are perpendicular to the same line, then the two lines are parallel to each other. � � � � � � SPIRAL REVIEW 39. A soccer league has 6 teams. During one season, each team plays each of the other teams 2 times. What is the total number of games played in the league during one season? (Previous course) Find the measure of each angle. (Lesson 1-4) 40. the supplement of ∠DJE 41. the complement of ∠FJG 42. the supplement of ∠GJH For the given information, name the postulate or theorem that proves ℓ ǁ m. (Lesson 3-3) 43. ∠2 ≅ ∠7 44. ∠3 ≅ ∠6 45. m∠4 + m∠6 = 180° � � �� 178 178 Chapter 3 Parallel and Perpendicular Lines 3-4 Use with Lesson 3-4 Activity Construct Perpendicular Lines In Lesson 3-4, you learned to construct the perpendicular bisector of a segment. This is the basis of the construction of a line perpendicular to a given line through a given point. The
steps in the construction are the same whether the point is on or off the line. TEKS G.2.A Geometric structure: use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. Also G.9.A Copy the given line ℓ and point P. 1 Place the compass point on P and draw an arc that intersects ℓ at two points. Label the points A and B. 2 Construct the perpendicular bisector of ̶̶ AB. Try This Copy each diagram and construct a line perpendicular to line ℓ through point P. Use a protractor to verify that the lines are perpendicular. 1. 2. 3. Follow the steps below to construct two parallel lines. Explain why ℓ ǁ n. Step 1 Given a line ℓ, draw a point P not on ℓ. Step 2 Construct line m perpendicular to ℓ through P. Step 3 Construct line n perpendicular to m through P. 3- 4 Geometry Lab 179 179 �� SECTION 3A Parallel and Perpendicular Lines and Transversals On the Spot Inside a mystery spot building, objects can appear to roll uphill, and people can look as if they are standing at impossible angles. This is because there is no view of the outside, so the room appears to be normal. Suppose that the ground is perfectly level and the floor of the building forms a 25° angle with the ground. The floor and ceiling are parallel, and the walls are perpendicular to the floor. 1. A table is placed in the room. The legs of the table are perpendicular to the floor, and the top is perpendicular to the legs. Draw a diagram and describe the relationship of the tabletop to the floor, walls, and ceiling of the room. 2. Find the angle of the table top relative to the ground. Suppose a ball is placed on the table. Describe what would happen and how it would appear to a person in the room. 3. Two people of the same height are standing on opposite ends of a board that makes a 25° angle with the floor, as shown. Explain how you know that the board is parallel to the ground. What would appear to be happening from the point of view of a person inside the room? 4. In the room, a lamp hangs from the ceiling along a line perpendicular to the ground. Find the angle the line makes with the walls. Describe how it would appear to a person standing in the room.
180 180 Chapter 3 Parallel and Perpendicular Lines �� Quiz for Lessons 3-1 Through 3-4 SECTION 3A 3-1 Lines and Angles Identify each of the following. 1. a pair of perpendicular segments 2. a pair of skew segments 3. a pair of parallel segments 4. a pair of parallel planes Give an example of each angle pair. 5. alternate interior angles 6. alternate exterior angles 7. corresponding angles 8. same-side interior angles 3-2 Angles Formed by Parallel Lines and Transversals Find each angle measure. 9. 10. 11. 3-3 Proving Lines Parallel Use the given information and the theorems and postulates you have learned to show that a ǁ b. 12. m∠8 = (13x + 20) °, m∠6 = (7x + 38) °, x = 3 13. ∠1 ≅ ∠5 14. m∠8 + m∠7 = 180° 15. m∠8 = m∠4 16. The tower shown is supported by guy wires such that m∠1 = (3x + 12) °, m∠2 = (4x - 2) °, and x = 14. Show that the guy wires are parallel. 3-4 Perpendicular Lines 17. Write a two-column proof. Given: ∠1 ≅ ∠2, ℓ ⊥ n Prove: ℓ ⊥ p Ready to Go On? 181 181 �� 3-5 Slopes of Lines TEKS G.7.B Dimensionality and the geometry of location: use slopes... to investigate geometric relationships, including parallel lines, perpendicular lines,.... Also G.7.A, G.7.C Objectives Find the slope of a line. Use slopes to identify parallel and perpendicular lines. Vocabulary rise run slope Why learn this? You can use the graph of a line to describe your rate of change, or speed, when traveling. (See Example 2.) The slope of a line in a coordinate plane is a number that describes the steepness of the line. Any two points on a line can be used to determine the slope. Slope of a Line DEFINITION EXAMPLE The rise is the difference in the y-values of two points on
a line. The run is the difference in the x-values of two points on a line. The slope of a line is the ratio of rise to run. If ( x 1, y 1 ) and ( x 2, y 2 ) are any two points on line, the slope of the line is Finding the Slope of a Line Use the slope formula to determine the slope of each line. A    AB B    CD A fraction with zero in the denominator is undefined because it is impossible to divide by zero. Substitute (2, 3) for ( x 1, y 1 ) and (7, 5) for ( x 2, y 2 ) in the slope formula and then simplify Substitute (4, -3) for ( x 1, y 1 ) and (4, 5) for ( x 2, y 2 ) in the slope formula and then simplify. 5 - (-3 The slope is undefined. = 8 _ 0 m = 182 182 Chapter 3 Parallel and Perpendicular Lines �� Use the slope formula to determine the slope of each line. C   EF D  GH Substitute (3, 4) for ( x 1, y 1 ) and (6, 4) for ( x 2, y 2 ) in the slope formula and then simplify0 Substitute (6, 2) for ( x 1, y 1 ) and (2, 6) for ( x 2, y 2 ) in the slope formula and then simplify4 = -1 1. Use the slope formula to determine the slope of   JK through J (3, 1) and K (2, -1). Positive Slope Negative Slope Zero Slope Undefined Slope Summary: Slope of a Line One interpretation of slope is a rate of change. If y represents miles traveled and x represents time in hours, the slope gives the rate of change in miles per hour. E X A M P L E 2 Transportation Application Transportation A trip from Dallas to Atlanta, Georgia covers 781 miles and crosses three states but is still shorter than a trip across Texas. The distance from Orange, Texas to El Paso via Interstate 10 is 859 miles. Tony is driving from Dallas, Texas,
to Atlanta, Georgia. At 3:00 P.M., he is 180 miles from Dallas. At 5:30 P.M., he is 330 miles from Dallas. Graph the line that represents Tony’s distance from Dallas at a given time. Find and interpret the slope of the line. Use the points (3, 180) and (5.5, 330) to graph the line and find the slope. m = 330 - 180 _ 5.5 - 3 = 150 _ 2.5 = 60 The slope is 60, which means he is traveling at an average speed of 60 miles per hour. 2. What if…? Use the graph above to estimate how far Tony will have traveled by 6:30 P.M. if his average speed stays the same. 3- 5 Slopes of Lines 183 183 �� Slopes of Parallel and Perpendicular Lines 3-5-1 Parallel Lines Theorem In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel. 3-5-2 Perpendicular Lines Theorem In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1. Vertical and horizontal lines are perpendicular. If a line has a slope of a_ b and - b _ a are called opposite reciprocals. The ratios a _ b, then the slope of a perpendicular line is - b_ a. E X A M P L E 3 Determining Whether Lines Are Parallel, Perpendicular, or Neither Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. A B C Four given points do not always determine two lines. Graph the lines to make sure the points are not collinear.    AB and    CD for A (2, 1), B (1, 5), C (4, 2), and D (5, -2) slope of   AB = 5 - 1 _ = -4 1 - 2 = 4 _ -1 = -4 _ 1 slope of   CD = -2 - 2 _ 5 - 4 The lines have the same slope, so they are
parallel. = -4    ST and    UV for S (-2, 2), T (5, -1), U (3, 4), and V (-1, -4) slope of   ST = -1 - 2 _ 5 - (-2) slope of   UV = -4 - 4 _ -1 - 3 = -8 _ -4 = 2 The slopes are not the same, so the lines are not parallel. The product of the slopes is not -1, so the lines are not perpendicular.    FG and    HJ for F (1, 1), G (2, 2), H (2, 1), and J (1, 2) slope of   FG = 2 - 1 _ = 1 slope of   HJ = 2 - 1 _ = -1 = 1 _ 1 = 1 _ -1 2 - 1 1 - 2 The product of the slopes is 1 (-1) = -1, so the lines are perpendicular. Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. 3a.   WX and   YZ for W (3, 1), X (3, -2), Y (-2, 3), and Z (4, 3) 3b.   KL and   MN for K (-4, 4), L (-2, -3), M (3, 1), and N (-5, -1) 3c.   BC and   DE for B (1, 1), C (3, 5), D (-2, -6), and E (3, 4) 184 184 Chapter 3 Parallel and Perpendicular Lines �� THINK AND DISCUSS 1. Explain how to find the slope of a line when given two points. 2. Compare the slopes of horizontal and vertical
lines. 3. GET ORGANIZED Copy and complete the graphic organizer. 3-5 Exercises Exercises KEYWORD: MG7 3-5 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary The slope of a line is the ratio of its? to its ̶̶̶?. (rise or run) ̶̶̶ Use the slope formula to determine the slope of each line. p. 182 2.  MN 3.   CD 4.  AB 5.   ST. 183 6. Biology A migrating bird flying at a constant speed travels 80 miles by 8:00 A.M. and 200 miles by 11:00 A.M. Graph the line that represents the bird’s distance traveled. Find and interpret the slope of the line. 184 Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. 7.   HJ and   KM for H (3, 2), J (4, 1), K (-2, -4), and M (-1, -5) 8.   LM and   NP for L (-2, 2), M (2, 5), N (0, 2), and P (3, -2) 9.   QR and   ST for Q (6, 1), R (-2, 4), S (5, 3), and T (-3, -1) 3- 5 Slopes of Lines 185 185 �� Independent Practice Use the slope formula to determine the slope of each line. PRACTICE AND PROBLEM SOLVING For See Exercises Example 10.  AB 11.   CD 10–13 14 15–17 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S9 Application Practice p. S30
12.  EF 13.   GH 14. Aviation A pilot traveling at a constant speed flies 100 miles by 2:30 P.M. and 475 miles by 5:00 P.M. Graph the line that represents the pilot’s distance flown. Find and interpret the slope of the line. Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. 15.  AB and   CD for A (2, -1), B (7, 2), C (2, -3), and D (-3, -6) 16.  XY and   ZW for X (-2, 5), Y (6, -2), Z (-3, 6), and W (4, 0) 17.  JK and   JL for J (-4, -2), K (4, -2), and L (-4, 6) 18. Geography The Rio Grande has an elevation of about 1150 meters above sea level in El Paso. The length of the river from that point to Brownsville where it enters the sea is about 2400 km. Find and interpret the slope of the river. For F (7, 6), G (-3, 5), H (-2, -3), J (4, -2), and K (6, 1), find each slope. 19.   FG 20.   GJ 21.  HK 22.   GK 23. Critical Thinking The slope of AB is greater than 0 and less than 1. Write an inequality for the slope of a line perpendicular to   AB. 24. Write About It Two cars are driving at the same speed. What is true about the lines that represent the distance traveled by each car at a given time? 25. This problem will prepare you for the Multi-Step TAKS Prep on page 200. A traffic engineer calculates the
speed of vehicles as they pass a traffic light. While the light is green, a taxi passes at a constant speed. After 2 s the taxi is 132 ft past the light. After 5 s it is 330 ft past the light. a. Find the speed of the taxi in feet per second. b. Use the fact that 22 ft/s = 15 mi/h to find the taxi’s speed in miles per hour. 186 186 Chapter 3 Parallel and Perpendicular Lines �� 26.   AB ⊥   CD for A (1, 3), B (4, -2), C (6, 1), and D (x, y). Which are possible values of x and y? x = 1, y = -2 x = 3, y = 6 x = 3, y = -4 x = -2, y = -4 27. Classify   MN and   PQ for M (-3, 1), N (1, 3), P (8, 4), and Q (2, 1). Parallel Perpendicular Vertical Skew 28. In the formula d = rt, d represents distance, and r represents the rate of change, or slope. Which ray on the graph represents a slope of 45 miles per hour? A B C D CHALLENGE AND EXTEND Use the given information to classify   29. a = c JK for J (a, b) and K (c, d). 30. b = d 31. The vertices of square ABCD are A (0, -2), B (6, 4), C (0, 10), D (-6, 4). a. Show that the opposite sides are parallel. b. Show that the consecutive sides are perpendicular. c. Show that all sides are congruent. 32.   ST ǁ   VW for S (-3, 5), T (1, -1), V (x, -3), and W (1, y). Find a set of possible values for x and y. 33. �

divisible by 2. Conjecture: If a number is divisible by 4, then it is even. Let x, y, and z represent the following. x : A number is divisible by 4. y : A number is divisible by 2. z : A number is even. You are given that x → y and z → y. The Law of Syllogism cannot be used to draw a conclusion since y is the conclusion of both conditionals. Even though the conjecture x → z is true, the logic used to draw the conclusion is not valid. 3. Determine if the conjecture is valid by the Law of Syllogism. Given: If an animal is a mammal, then it has hair. If an animal is a dog, then it is a mammal. Conjecture: If an animal is a dog, then it has hair. E X A M P L E 4 Applying the Laws of Deductive Reasoning Draw a conclusion from the given information. A Given: If a team wins 10 games, then they play in the finals. If a team plays in the finals, then they travel to Boston. The Ravens won 10 games. Conclusion: The Ravens will travel to Boston. B Given: If two angles form a linear pair, then they are adjacent. If two angles are adjacent, then they share a side. ∠1 and ∠2 form a linear pair. Conclusion: ∠1 and ∠2 share a side. 4. Draw a conclusion from the given information. Given: If a polygon is a triangle, then it has three sides. If a polygon has three sides, then it is not a quadrilateral. Polygon P is a triangle. THINK AND DISCUSS 1. Could “A square has exactly two sides” be the conclusion of a valid argument? If so, what do you know about the truth value of the given information? 2. Explain why writing conditional statements as symbols might help you evaluate the validity of an argument. 3. GET ORGANIZED Copy and complete the graphic organizer. Write each law in your own words and give an example of each. 90 90 Chapter 2 Geometric Reasoning �� 2-3 Exercises Exercises KEYWORD: MG7 2-3 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Explain how deductive reasoning differs from inductive reasoning Does each conclusion

use inductive or deductive reasoning? p. 88 2. At Bell High School, students must take Biology before they take Chemistry. Sam is in Chemistry, so Marcia concludes that he has taken Biology. 3. A detective learns that his main suspect was out of town the day of the crime. He concludes that the suspect is innocent Determine if each conjecture is valid by the Law of Detachment. p. 89 4. Given: If you want to go on a field trip, you must have a signed permission slip. Zola has a signed permission slip. Conjecture: Zola wants to go on a field trip. 5. Given: If the side lengths of a rectangle are 3 ft and 4 ft, then its area is 12 ft 2. A rectangle has side lengths of 3 ft and 4 ft. Conjecture: The area of the rectangle is 12 ft Determine if each conjecture is valid by the Law of Syllogism. p. 89 6. Given: If you fly from Texas to California, you travel from the central to the Pacific time zone. If you travel from the central to the Pacific time zone, then you gain two hours. Conjecture: If you fly from Texas to California, you gain two hours. 7. Given: If a figure is a square, then the figure is a rectangle. If a figure is a square, then it is a parallelogram. Conjecture: If a figure is a parallelogram, then it is a rectangle. 90 8. Draw a conclusion from the given information. Given: If you leave your car lights on overnight, then your car battery will drain. If your battery is drained, your car might not start. Alex left his car lights on last night. Independent Practice Does each conclusion use inductive or deductive reasoning? PRACTICE AND PROBLEM SOLVING For See Exercises Example 9–10 11 12 13 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S6 Application Practice p. S29 9. The sum of the angle measures of a triangle is 180°. Two angles of a triangle measure 40° and 60°, so Kandy concludes that the third angle measures 80°. 10. All of the students in Henry’s Geometry class are juniors. Alexander takes Geometry, but has another teacher. Henry concludes that Alexander is also a junior. 11. Determine if the conjecture is valid by the Law of Detachment

. Given: If one integer is odd and another integer is even, their product is even. The product of two integers is 24. Conjecture: One of the two integers is odd. 2- 3 Using Deductive Reasoning to Verify Conjectures 91 91 � 12. Science Determine if the conjecture is valid by the Law of Syllogism. Given: If an element is an alkali metal, then it reacts with water. If an element is in the first column of the periodic table, then it is an alkali metal. Conjecture: If an element is in the first column of the periodic table, then it reacts with water. 13. Draw a conclusion from the given information. Given: If Dakota watches the news, she is informed about current events. If Dakota knows about current events, she gets better grades in Social Studies. Dakota watches the news. 14. Technology Joseph downloads a file in 18 minutes with a dial-up modem. How long would it take to download the file with a Cheetah-Net cable modem? Recreation Recreation Use the true statements below for Exercises 15–18. Determine whether each conclusion is valid. The Top Thrill Dragster is 420 feet tall and includes a 400-foot vertical drop. It twists 270° as it drops. It is one of 16 roller coasters at Cedar Point amusement park. I. The Top Thrill Dragster is at Cedar Point amusement park in Sandusky, OH. II. Carter and Mary go to Cedar Point. III. The Top Thrill Dragster roller coaster reaches speeds of 120 mi/h. IV. When Carter goes to an amusement park, he rides all the roller coasters. 15. Carter went to Sandusky, OH. 16. Mary rode the Top Thrill Dragster. 17. Carter rode a roller coaster that travels 120 mi/h. 18. Mary rode a roller coaster that travels 120 mi/h. 19. Critical Thinking Is the argument below a valid application of the Law of Syllogism? Is the conclusion true? Explain your answers. If 3 - x < 5, then x < -2. If x < -2, then -5x > 10. Thus, if 3 - x < 5, then -5x > 10. 20. /////ERROR ANALYSIS///// Below are two conclusions. Which is incorrect? Explain the error. If two angles are complementary, their measures add to 90°. If an angle measures 90°,

then it is a right angle. ∠A and ∠B are complementary. 21. Write About It Write one example of a real-life logical argument that uses the Law of Detachment and one that uses the Law of Syllogism. Explain why the conclusions are valid. 22. This problem will prepare you for the Multi-Step TAKS Prep on page 102. When Alice meets the Pigeon in Wonderland, the Pigeon thinks she is a serpent. The Pigeon reasons that serpents eat eggs, and Alice confirms that she has eaten eggs. a. Write “Serpents eat eggs” as a conditional statement. b. Is the Pigeon’s conclusion that Alice is a serpent valid? Explain your reasoning. 92 92 Chapter 2 Geometric Reasoning �� 23. The Supershots scored over 75 points in each of ten straight games. The newspaper predicts that they will score more than 75 points tonight. Which form of reasoning is this conclusion based on? Deductive reasoning, because the conclusion is based on logic Deductive reasoning, because the conclusion is based on a pattern Inductive reasoning, because the conclusion is based on logic Inductive reasoning, because the conclusion is based on a pattern 24.  HF bisects ∠EHG. Which conclusion is NOT valid? E, F, and G are coplanar. ∠EHF ≅ ∠FHG ̶̶ EF ≅ ̶̶ FG m∠EHF = m∠FHG 25. Gridded Response If Whitney plays a low G on her piano, the frequency of the note is 24.50 hertz. The frequency of a note doubles with each octave. What is the frequency in hertz of a G note that is 3 octaves above low G? CHALLENGE AND EXTEND 26. Political Science To be eligible to hold the office of the president of the United States, a person must be at least 35 years old, be a natural-born U.S. citizen, and have been a U.S. resident for at least 14 years. Given this information, what conclusion, if any, can be drawn from the statements below? Explain your reasoning. Andre is not eligible to be the president of the United States. Andre has lived in the United States for 16 years. 27

. Multi-Step Consider the two conditional statements below. If you live in San Diego, then you live in California. If you live in California, then you live in the United States. a. Draw a conclusion from the given conditional statements. b. Write the contrapositive of each conditional statement. c. Draw a conclusion from the two contrapositives. d. How does the conclusion in part a relate to the conclusion in part c? 28. If Cassie goes to the skate park, Hanna and Amy will go. If Hanna or Amy goes to the skate park, then Marc will go. If Marc goes to the skate park, then Dallas will go. If only two of the five people went to the skate park, who were they? SPIRAL REVIEW Simplify each expression. (Previous course) 29. 2 (x + 5) 30. (4y + 6) - (3y - 5) 31. (3c + 4c) + 2 (-7c + 7) Find the coordinates of the midpoint of the segment connecting each pair of points. (Lesson 1-6) 32. (1, 2) and (4, 5) 33. (-3, 6) and (0, 1) 34. (-2.5, 9) and (2.5, -3) Identify the hypothesis and conclusion of each conditional statement. (Lesson 2-2) 35. If the fire alarm rings, then everyone should exit the building. 36. If two different lines intersect, then they intersect at exactly one point. 37. The statement ̶̶ AB ≅ ̶̶ CD implies that AB = CD. 2- 3 Using Deductive Reasoning to Verify Conjectures 93 93 �� 2-3 Solve Logic Puzzles In Lesson 2-3, you used deductive reasoning to analyze the truth values of conditional statements. Now you will learn some methods for diagramming conditional statements to help you solve logic puzzles. Use with Lesson 2-3 TEKS G.4.A Geometric structure: select an appropriate representation... in order to solve problems Activity 1 Bonnie, Cally, Daphne, and Fiona own a bird, cat, dog, and fish. No girl has a type of pet that begins with the same letter as her name. Bonnie is allergic to animal fur. Daphne feeds Fiona’s bird when Fiona is away. Make a table to determine who owns which animal. 1 Since no girl has a type of

pet that starts with the same letter as her name, place an X in each box along the diagonal of the table. 2 Bonnie cannot have a cat or dog because of her allergy. So she must own the fish, and no other girl can have the fish. Bird Cat Dog Fish Bird Cat Dog Fish Bonnie × Cally Daphne Fiona × × × Bonnie × Cally Daphne Fiona × × × × ✓ × × × 3 Fiona owns the bird, so place a check in 4 Therefore, Daphne owns the cat, and Cally Fiona’s row, in the bird column. Place an X in the remaining boxes in the same column and row. owns the dog. Bird Cat Dog Fish Bird Cat Dog Fish Bonnie Cally Daphne Fiona × × × ✓ × × × × × × ✓ × × × Bonnie Cally Daphne Fiona × × × ✓ × × ✓ × × ✓ × × ✓ × × × Try This 1. After figuring out that Fiona owns the bird in Step 3, why can you place an X in every other box in that row and column? 2. Ally, Emily, Misha, and Tracy go to a dance with Danny, Frank, Jude, and Kian. Ally and Frank are siblings. Jude and Kian are roommates. Misha does not know Kian. Emily goes with Kian’s roommate. Tracy goes with Ally’s brother. Who went to the dance with whom? Ally Emily Misha Tracy 94 94 Chapter 2 Geometric Reasoning Danny Frank Jude Kian Activity 2 A farmer has a goat, a wolf, and a cabbage. He wants to transport all three from one side of a river to the other. He has a boat, but it has only enough room for the farmer and one thing. The wolf will eat the goat if they are left alone together, and the goat will eat the cabbage if they are left alone. How can the farmer get everything to the other side of the river? You can use a network to solve this kind of puzzle. A network is a diagram of vertices and edges, also known as a graph. An edge is a curve or a segment that joins two vertices of the graph. A vertex is a point on the graph. 1 Let F represent the farmer, W represent the wolf, G represent the goat, and C represent the cabbage. Use an ordered pair to represent what is on each side of the river. The first ordered pair is (FWGC, —),

and the desired result is (—, FWGC). 2 Draw a vertex and label it with the first ordered pair. Then draw an edge and vertex for each possible trip the farmer could make across the river. If at any point a path results in an unworkable combination of things, no more edges can be drawn from that vertex. 3 From each workable vertex, continue to draw edges and vertices that represent the next trip across the river. When you get to a vertex for (—, FWGC), the network is complete. 4 Use the network to write out the solution in words. Try This 3. What combinations are unworkable? Why? 4. How many solutions are there to the farmer’s transport problem? How many steps does each solution take? 5. What is the advantage of drawing a complete solution network rather than working out one solution with a diagram? 6. Madeline has two measuring cups—a 1-cup measuring cup and a 3__ 4 -cup measuring cup. Neither cup has any markings on it. How can Madeline get exactly 1 __ 2 cup of flour in the larger measuring cup? Complete the network below to solve the problem. 2- 3 Geometry Lab 95 95 �� 2-4 Biconditional Statements and Definitions TEKS G.3.A Geometric structure: determine the validity of a conditional statement, its converse, inverse, and contrapositive. Also G.3.B Objective Write and analyze biconditional statements. Vocabulary biconditional statement definition polygon triangle quadrilateral Who uses this? A gardener can plan the color of the hydrangeas she plants by checking the pH of the soil. The pH of a solution is a measure of the concentration of hydronium ions in the solution. If a solution has a pH less than 7, it is an acid. Also, if a solution is an acid, it has a pH less than 7. When you combine a conditional statement and its converse, you create a biconditional statement. A biconditional statement is a statement that can be written in the form “p if and only if q.” This means “if p, then q” and “if q, then p.” The biconditional “p

if and only if q” can also be written as “p iff q” or p ↔ q. So you can define an acid with the following biconditional statement: A solution is an acid if and only if it has a pH less than 7. E X A M P L E 1 Identifying the Conditionals within a Biconditional Statement Write the conditional statement and converse within each biconditional. A Two angles are congruent if and only if their measures are equal. Let p and q represent the following. p : Two angles are congruent. q : Two angle measures are equal. The two parts of the biconditional p ↔ q are p → q and q → p. Conditional: If two angles are congruent, then their measures are equal. Converse: If two angle measures are equal, then the angles are congruent. B A solution is a base ↔ it has a pH greater than 7. Let x and y represent the following. x : A solution is a base. y : A solution has a pH greater than 7. The two parts of the biconditional x ↔ y are x → y and y → x. Conditional: If a solution is a base, then it has a pH greater than 7. Converse: If a solution has a pH greater than 7, then it is a base. Write the conditional statement and converse within each biconditional. 1a. An angle is acute iff its measure is greater than 0° and less than 90°. 1b. Cho is a member if and only if he has paid the $5 dues. 96 96 Chapter 2 Geometric Reasoning ��meansandProject TitleGeometry 2007 Student EditionSpec Numberge07sec01l04002aCreated ByKrosscore CorporationCreation Date10/07/2004 E X A M P L E 2 Writing a Biconditional Statement For each conditional, write the converse and a biconditional statement. A If 2x + 5 = 11, then x = 3. Converse: If x = 3, then 2x + 5 = 11. Biconditional: 2x + 5 = 11 if and only if x = 3. B If a point is a midpoint, then it divides the segment into two congruent segments. Converse: If a point divides a segment into two congruent segments

, then the point is a midpoint. Biconditional: A point is a midpoint if and only if it divides the segment into two congruent segments. For each conditional, write the converse and a biconditional statement. 2a. If the date is July 4th, then it is Independence Day. 2b. If points lie on the same line, then they are collinear. For a biconditional statement to be true, both the conditional statement and its converse must be true. If either the conditional or the converse is false, then the biconditional statement is false. E X A M P L E 3 Analyzing the Truth Value of a Biconditional Statement Determine if each biconditional is true. If false, give a counterexample. A A square has a side length of 5 if and only if it has an area of 25. Conditional: If a square has a side length of 5, then it has an area of 25. The conditional is true. Converse: If a square has an area of 25, then it has a side length of 5. The converse is true. Since the conditional and its converse are true, the biconditional is true. B The number n is a positive integer ↔ 2n is a natural number. Conditional: If n is a positive integer, then 2n is a natural number. The conditional is true. Converse: If 2n is a natural number, then n is a positive integer. The converse is false. If 2n = 1, then n = 1 __ 2, which is not an integer. Because the converse is false, the biconditional is false. Determine if each biconditional is true. If false, give a counterexample. 3a. An angle is a right angle iff its measure is 90°. 3b. y = -5 ↔ y 2 = 25 In geometry, biconditional statements are used to write definitions. A definition is a statement that describes a mathematical object and can be written as a true biconditional. Most definitions in the glossary are not written as biconditional statements, but they can be. The “if and only if” is implied. 2- 4 Biconditional Statements and Definitions 97 97 In the glossary, a polygon is defined as a closed plane figure formed by three or

more line segments. Each segment intersects exactly two other segments only at their endpoints, and no two segments with a common endpoint are collinear. Polygons Not Polygons A triangle is defined as a three-sided polygon, and a quadrilateral is a four-sided polygon. A good, precise definition can be used forward and backward. For example, if a figure is a quadrilateral, then it is a four-sided polygon. If a figure is a four-sided polygon, then it is a quadrilateral. To make sure a definition is precise, it helps to write it as a biconditional statement. E X A M P L E 4 Writing Definitions as Biconditional Statements Write each definition as a biconditional. A A triangle is a three-sided polygon. Think of definitions as being reversible. Postulates, however, are not necessarily true when reversed. A figure is a triangle if and only if it is a three-sided polygon. B A segment bisector is a ray, segment, or line that divides a segment into two congruent segments. A ray, segment, or line is a segment bisector if and only if it divides a segment into two congruent segments. Write each definition as a biconditional. 4a. A quadrilateral is a four-sided polygon. 4b. The measure of a straight angle is 180°. THINK AND DISCUSS 1. How do you determine if a biconditional statement is true or false? 2. Compare a triangle and a quadrilateral. 3. GET ORGANIZED Copy and complete the graphic organizer. Use the definition of a polygon to write a conditional, converse, and biconditional in the appropriate boxes. 98 98 Chapter 2 Geometric Reasoning �� 2-4 Exercises Exercises KEYWORD: MG7 2-4 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary How is a biconditional statement different from a conditional statement Write the conditional statement and converse within each biconditional. p. 96 2. Perry can paint the entire living room if and only if he has enough paint. 3. Your medicine will be ready by 5 P.M. if and only if you drop your prescription off by 8 A.M

For each conditional, write the converse and a biconditional statement. p. 97 4. If a student is a sophomore, then the student is in the tenth grade. 5. If two segments have the same length, then they are congruent. 97 Multi-Step Determine if each biconditional is true. If false, give a counterexample. 6. xy = 0 ↔ x = 0 or y = 0. 7. A figure is a quadrilateral if and only if it is a polygon Write each definition as a biconditional. p. 98 8. Parallel lines are two coplanar lines that never intersect. 9. A hummingbird is a tiny, brightly colored bird with narrow wings, a slender bill, and a long tongue. Independent Practice Write the conditional statement and converse within each biconditional. PRACTICE AND PROBLEM SOLVING For See Exercises Example 10–12 13–15 16–17 18–19 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S6 Application Practice p. S29 10. Three points are coplanar if and only if they lie in the same plane. 11. A parallelogram is a rectangle if and only if it has four right angles. 12. A lunar eclipse occurs if and only if Earth is between the Sun and the Moon. For each conditional, write the converse and a biconditional statement. 13. If today is Saturday or Sunday, then it is the weekend. 14. If Greg has the fastest time, then he wins the race. 15. If a triangle contains a right angle, then it is a right triangle. Multi-Step Determine if each biconditional is true. If false, give a counterexample. 16. Felipe is a swimmer if and only if he is an athlete. 17. The number 2n is even if and only if n is an integer. Write each definition as a biconditional. 18. A circle is the set of all points in a plane that are a fixed distance from a given point. 19. A catcher is a baseball player who is positioned behind home plate and who catches throws from the pitcher. 2- 4 Biconditional Statements and Definitions 99 99 Algebra Determine if a true biconditional can be written from each conditional statement. If not, give a counterexample. 20

. If a = b, then ⎜a⎟ = ⎜b⎟. x + 8 = 12. 21. If 3x - 2 = 13, then 4 _ 5 23. If x > 0, then x 2 > 0. 22. If y 2 = 64, then 3y = 24. Use the diagrams to write a definition for each figure. 24. 25. Biology 26. Biology White blood cells are cells that defend the body against invading organisms by engulfing them or by releasing chemicals called antibodies. Write the definition of a white blood cell as a biconditional statement. White blood cells live less than a few weeks. A drop of blood can contain anywhere from 7000 to 25,000 white blood cells. Explain why the given statement is not a definition. 27. An automobile is a vehicle that moves along the ground. 28. A calculator is a machine that performs computations with numbers. 29. An angle is a geometric object formed by two rays. Chemistry Use the table for Exercises 30–32. Determine if a true biconditional statement can be written from each conditional. 30. If a solution has a pH of 4, then it is tomato juice. 31. If a solution is bleach, then its pH is 13. 32. If a solution has a pH greater than 7, then it is not battery acid. pH 0 4 6 8 13 14 Examples Battery Acid Acid rain, tomato juice Saliva Sea water Bleach, oven cleaner Drain cleaner Complete each statement to form a true biconditional. 33. The circumference of a circle is 10π if and only if its radius is?. ̶̶̶ 34. Four points in a plane form a? if and only if no three of them are collinear. ̶̶̶ 35. Critical Thinking Write the definition of a biconditional statement as a biconditional statement. Use the conditional and converse within the statement to explain why your biconditional is true. 36. Write About It Use the definition of an angle bisector to explain what is meant by the statement “A good definition is reversible.” 37. This problem will prepare you for the Multi-Step TAKS Prep on page 102. a. Write “I say what I mean” and “I mean what I say” as conditionals. b. Explain why the biconditional statement implied by Alice is false. “Then you should

say what you mean,” the March Hare went on. “I do,” Alice hastily replied; “at least—at least I mean what I say—that’s the same thing, you know.” 100 100 Chapter 2 Geometric Reasoning �� 38. Which is a counterexample for the biconditional “An angle measures 80° if and only if the angle is acute”? m∠S = 60° m∠S = 115° m∠S = 90° m∠S = 360° 39. Which biconditional is equivalent to the spelling phrase “I before E except after C”? The letter I comes before E if and only if I follows C. The letter E comes before I if and only if E follows C. The letter E comes before I if and only if E comes before C. The letter I comes before E if and only if I comes before C. 40. Which conditional statement can be used to write a true biconditional? If a number is divisible by 4, then it is even. If a ratio compares two quantities measured in different units, the ratio is a rate. If two angles are supplementary, then they are adjacent. If an angle is right, then it is not acute. 41. Short Response Write the two conditional statements that make up the biconditional “You will get a traffic ticket if and only if you are speeding.” Is the biconditional true or false? Explain your answer. CHALLENGE AND EXTEND 42. Critical Thinking Describe what the Venn diagram of a true biconditional statement looks like. How does this support the idea that a definition can be written as a true biconditional? 43. Consider the conditional “If an angle measures 105°, then the angle is obtuse.” a. Write the inverse of the conditional statement. b. Write the converse of the inverse. c. How is the converse of the inverse related to the original conditional? d. What is the truth value of the biconditional statement formed by the inverse of the original conditional and the converse of the inverse? Explain. 44. Suppose A, B, C, and D are coplanar, and A, B, and C are not collinear. What is the truth

value of the biconditional formed from the true conditional “If m∠ABD + m∠DBC = m∠ABC, then D is in the interior of ∠ABC”? Explain. 45. Find a counterexample for “n is divisible by 4 if and only if n 2 is even.” SPIRAL REVIEW Describe how the graph of each function differs from the graph of the parent function y = x 2. (Previous course) 46 47. y = -2 x 2 - 1 48. y = (x - 2) (x + 2) A transformation maps S onto T and X onto Y. Name each of the following. (Lesson 1-7) 49. the image of S 50. the image of X 51. the preimage of T Determine if each conjecture is true. If not, give a counterexample. (Lesson 2-1) 52. If n ≥ 0, then n _ 2 54. The vertices of the image of a figure under the translation (x, y) → (x + 0, y + 0) 53. If x is prime, then x + 2 is also prime. > 0. have the same coordinates as the preimage. 2- 4 Biconditional Statements and Definitions 101 101 SECTION 2A Inductive and Deductive Reasoning Rhyme or Reason Alice’s Adventures in Wonderland originated as a story told by Charles Lutwidge Dodgson (Lewis Carroll) to three young traveling companions. The story is famous for its wordplay and logical absurdities. 1. When Alice first meets the Cheshire Cat, she asks what sort of people live in Wonderland. The Cat explains that everyone in Wonderland is mad. What conjecture might the Cat make since Alice, too, is in Wonderland? 2. “I don’t much care where—” said Alice. “Then it doesn’t matter which way you go,” said the Cat. “—so long as I get somewhere,” Alice added as an explanation. “Oh, you’re sure to do that,” said the Cat, “if you only walk long enough.” Write the conditional statement implied by the Cat’s response to Alice. 3. “Well, then,” the Cat went on, “you see a dog growls when it’s

angry, and wags its tail when it’s pleased. Now I growl when I’m pleased, and wag my tail when I’m angry. Therefore I’m mad.” Is the Cat’s conclusion valid by the Law of Detachment or the Law of Syllogism? Explain your reasoning. 4. “You might just as well say,” added the Dormouse, who seemed to be talking in his sleep, “that ‘I breathe when I sleep’ is the same thing as ‘I sleep when I breathe’!” Write a biconditional statement from the Dormouse’s example. Explain why the biconditional statement is false. 102 102 Chapter 2 Geometric Reasoning Quiz for Lessons 2-1 Through 2-4 2-1 Using Inductive Reasoning to Make Conjectures Find the next item in each pattern. 1. 1, 10, 18, 25. July, May, March, … 3. 1 _ 4 2 8 5. A biologist recorded the following data about the weight of male lions in a wildlife park in Africa. Use the table to make a conjecture about the average weight of a male lion.,... 6. Complete the conjecture “The sum of two negative numbers is?.” ̶̶̶̶ 7. Show that the conjecture “If an even number is divided by 2, then the result is an even number” is false by finding a counterexample. SECTION 2A 4. ∣,,,... ID Number Weight (lb) A1902SM A1904SM A1920SM A1956SM A1974SM 387.2 420.5 440.6 398.7 415.0 2-2 Conditional Statements 8. Identify the hypothesis and conclusion of the conditional statement “An angle is obtuse if its measure is 107°.” Write a conditional statement from each of the following. 9. A whole number is 10. an integer. 11. The diagonals of a square are congruent. Determine if each conditional is true. If false, give a counterexample. 12. If an angle is acute, then it has a measure of 30°. 13. If 9x - 11 = 2x + 3, then x = 2. 14. Write the converse, inverse, and contrapositive of the statement “If

a number is even, then it is divisible by 4.” Find the truth value of each. 2-3 Using Deductive Reasoning to Verify Conjectures 15. Determine if the following conjecture is valid by the Law of Detachment. Given: If Sue finishes her science project, she can go to the movie. Sue goes to the movie. Conjecture: Sue finished her science project. 16. Use the Law of Syllogism to draw a conclusion from the given information. Given: If one angle of a triangle is 90°, then the triangle is a right triangle. If a triangle is a right triangle, then its acute angle measures are complementary. 2-4 Biconditional Statements and Definitions 17. For the conditional “If two angles are supplementary, the sum of their measures is 180°,” write the converse and a biconditional statement. 18. Determine if the biconditional “ √  x = 4 if and only if x = 16” is true. If false, give a counterexample. Ready to Go On? 103 103 �� 2-5 Algebraic Proof TEKS G.3.E Geometric structure: use deductive reasoning to prove a statement. Also G.3.B, G.3.C Objectives Review properties of equality and use them to write algebraic proofs. Identify properties of equality and congruence. Vocabulary proof The Distributive Property states that a (b + c) = ab + ac. Who uses this? Game designers and animators solve equations to simulate motion. (See Example 2.) A proof is an argument that uses logic, definitions, properties, and previously proven statements to show that a conclusion is true. If you’ve ever solved an equation in Algebra, then you’ve already done a proof! An algebraic proof uses algebraic properties such as the properties of equality and the Distributive Property. Properties of Equality Properties of Equality Addition Property of Equality If a = b, then a + c = b + c. Subtraction Property of Equality If a = b, then a - c = b - c. Multiplication Property of Equality Division Property of Equality Reflexive Property of Equality If a = b, then ac = bc. If a = b and c ≠ 0, then a __ c = b __ c. a = a Symmetric Property of Equality If a =

b, then b = a. Transitive Property of Equality If a = b and b = c, then a = c. Substitution Property of Equality If a = b, then b can be substituted for a in any expression. As you learned in Lesson 2-3, if you start with a true statement and each logical step is valid, then your conclusion is valid. An important part of writing a proof is giving justifications to show that every step is valid. For each justification, you can use a definition, postulate, property, or a piece of information that is given. E X A M P L E 1 Solving an Equation in Algebra Solve the equation -5 = 3n + 1. Write a justification for each step. -5 = 3n + 1 - 1 - 1 ̶̶̶̶̶ ̶̶̶ -6 = 3n _ _ -6 3n 3 3 -2 = n n = -2 = Given equation Subtraction Property of Equality Simplify. Division Property of Equality Simplify. Symmetric Property of Equality 1. Solve the equation 1 _ 2 t = -7. Write a justification for each step. 104 104 Chapter 2 Geometric Reasoning E X A M P L E 2 Problem-Solving Application To simulate the motion of an object in a computer game, the designer uses the formula sr = 3.6p to find the number of pixels the object must travel during each second of animation. In the formula, s is the desired speed of the object in kilometers per hour, r is the scale of pixels per meter, and p is the number of pixels traveled per second. The graphics in a game are based on a scale of 6 pixels per meter. The designer wants to simulate a vehicle moving at 75 km/h. How many pixels must the vehicle travel each second? Solve the equation for p and justify each step. Understand the Problem The answer will be the number of pixels traveled per second. List the important information: • sr = 3.6p • p: pixels traveled per second • s = 75 km/h • r = 6 pixels per meter Make a Plan Substitute the given information into the formula and solve. Solve sr = 3.6p (75) (6) = 3.6p 450 = 3.6p _ _ 3.6p 450 3.6 3.6 125 = p = Given equation Substitution Property of Equality Simplify. Division Property of Equality Simplify

. p = 125 pixels Symmetric Property of Equality Look Back Check your answer by substituting it back into the original formula. sr = 3.6p (75) (6) = 3.6 (125) 450 = 450 ✓ AB represents the ̶̶ length of AB, so you can think of AB as a variable representing a number. 2. What is the temperature in degrees Celsius C when it is 86°F? Solve the equation C = 5 _ (F - 32) for C and justify each step. 9 Like algebra, geometry also uses numbers, variables, and operations. For example, segment lengths and angle measures are numbers. So you can use these same properties of equality to write algebraic proofs in geometry. 2- 5 Algebraic Proof 105 105 1234�� E X A M P L E 3 Solving an Equation in Geometry Write a justification for each step. KM = KL + LM 5x - 4 = (x + 3) + (2x - 1) 5x - 4 = 3x + 2 2x - 4 = 2 2x = 6 x = 3 Segment Addition Postulate Substitution Property of Equality Simplify. Subtraction Property of Equality Addition Property of Equality Division Property of Equality 3. Write a justification for each step. m∠ABC = m∠ABD + m∠DBC 8x ° = (3x + 5) ° + (6x - 16) ° 8x = 9x - 11 -x = -11 x = 11 You learned in Chapter 1 that segments with equal lengths are congruent and that angles with equal measures are congruent. So the Reflexive, Symmetric, and Transitive Properties of Equality have corresponding properties of congruence. Properties of Congruence SYMBOLS EXAMPLE Reflexive Property of Congruence figure A ≅ figure A (Reflex. Prop. of ≅) Symmetric Property of Congruence ̶̶ EF ≅ ̶̶ EF If figure A ≅ figure B, then figure B ≅ figure A. (Sym. Prop. of ≅) If ∠1 ≅ ∠2, then ∠2 ≅ ∠1. Transitive Property of Congruence If figure A ≅ figure B and figure B ≅ figure C, then figure A ≅ figure C. (Trans. Prop. of ≅) ̶̶ PQ ≅

̶̶ PQ ≅ ̶̶ RS and ̶̶ TU. If then ̶̶ RS ≅ ̶̶ TU, E X A M P L E 4 Identifying Properties of Equality and Congruence Identify the property that justifies each statement. Numbers are equal (=) and figures are congruent (≅). A m∠1 = m∠1 B ̶̶ XY ≅ ̶̶ VW, so ̶̶ VW ≅ ̶̶ XY. C ∠ABC ≅ ∠ABC Reflex. Prop. of = Sym. Prop. of ≅ Reflex. Prop. of ≅ D ∠1 ≅ ∠2, and ∠2 ≅ ∠3. So ∠1 ≅ ∠3. Trans. Prop. of ≅ Identify the property that justifies each statement. 4a. DE = GH, so GH = DE. 4c. 0 = a, and a = x. So 0 = x. 4b. 94° = 94° 4d. ∠A ≅ ∠Y, so ∠Y ≅ ∠A. 106 106 Chapter 2 Geometric Reasoning �� THINK AND DISCUSS 1. Tell what property you would use to solve the equation k _ 6 2. Explain when to use a congruence symbol instead of an equal sign. = 3.5. 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, write an example of the property, using the correct symbol. 2-5 Exercises Exercises KEYWORD: MG7 2-5 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Write the definition of proof in your own words Multi-Step Solve each equation. Write a justification for each step. p. 104 2. y + 1 = 5 3. t - 3.2 = -8.3 4. 2p - 30 = -4p + 6 n = 3 _ 6. 1 _ 4 2 5. x + 3 _ -2 = 8 7. 0 = 2 (r - 3. Nutrition Amy’s favorite breakfast cereal has 102 Calories per serving. The equation p. 105 C = 9f + 90 relates the grams of fat f in one serving to the Calories C in one serving. How many grams of fat are in one serving of the

cereal? Solve the equation for f and justify each step. 9. Movie Rentals The equation C = $5.75 + $0.89m relates the number of movie rentals m to the monthly cost C of a movie club membership. How many movies did Elias rent this month if his membership cost $11.98? Solve the equation for m and justify each step Write a justification for each step. p. 106 10. 11. AB = BC 5y + 6 = 2y + 21 3y + 6 = 21 3y = 15 y = 5 PQ + QR = PR 3n + 25 = 9n -5 25 = 6n -5 30 = 6n. 106 12. ̶̶ AB ≅ ̶̶ AB Identify the property that justifies each statement. 14. x = y, so y = x. 13. m∠1 = m∠2, and m∠2 = m∠4. So m∠1 = m∠4. ̶̶ PR. So ̶̶ YZ, and ̶̶ YZ ≅ ̶̶ ST ≅ ̶̶ ST ≅ ̶̶ PR. 15. 2- 5 Algebraic Proof 107 107 �� Independent Practice Multi-Step Solve each equation. Write a justification for each step. PRACTICE AND PROBLEM SOLVING For See Exercises Example 16. 5x - 3 = 4 (x + 2) 17. 1.6 = 3.2n 19. - (h + 3) = 72 20. 9y + 17 = -19 - 2 = -10 18. z _ 3 21. 1 _ (p - 16) = 13 2 16–21 22 23–24 25–28 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S7 Application Practice p. S29 22. Ecology The equation T = 0.03c + 0.05b relates the numbers of cans c and bottles b collected in a recycling rally to the total dollars T raised. How many cans were collected if $147 was raised and 150 bottles were collected? Solve the equation for c and justify each step. Write a justification for each step. 23. m∠XYZ = m∠2 + m∠3 4n - 6 = 58 + (2n - 12

) 4n - 6 = 2n + 46 2n - 6 = 46 2n = 52 n = 26 24. m∠WYV = m∠1 + m∠2 5n = 3 (n - 2) + 58 5n = 3n - 6 + 58 5n = 3n + 52 2n = 52 n = 26 Identify the property that justifies each statement. 25. ̶̶ KL ≅ ̶̶ PR, so ̶̶ PR ≅ ̶̶ KL. 26. 412 = 412 27. If a = b and b = 0, then a = 0. 28. figure A ≅ figure A 29. Estimation Round the numbers in the equation 2 (3.1x - 0.87) = 94.36 to the nearest whole number and estimate the solution. Then solve the equation, justifying each step. Compare your estimate to the exact solution. Use the indicated property to complete each statement. 30. Reflexive Property of Equality: 3x - 1 =? ̶̶̶ 31. Transitive Property of Congruence: If ∠ A ≅ ∠ X and ∠ X ≅ ∠T, then 32. Symmetric Property of Congruence: If ̶̶ BC ≅ ̶̶ NP, then?. ̶̶̶?. ̶̶̶ 33. Recreation The north campground is midway between the Northpoint Overlook and the waterfall. Use the midpoint formula to find the values of x and y, and justify each step. 34. Business A computer repair technician charges $35 for each job plus $21 per hour of labor and 110% of the cost of parts. The total charge for a 3-hour job was $169.50. What was the cost of parts for this job? Write and solve an equation and justify each step in the solution. 35. Finance Morgan spent a total of $1,733.65 on her car last year. She spent $92.50 on registration, $79.96 on maintenance, and $983 on insurance. She spent the remaining money on gas. She drove a total of 10,820 miles. a. How much on average did the gas cost per mile? Write and solve an equation and justify each step in the solution. b. What if…? Suppose Morgan’s car averages 32 miles per gallon of gas. How much on average did Morgan pay for a gallon of gas? 36. Critical

Thinking Use the definition of segment congruence and the properties of equality to show that all three properties of congruence are true for segments. 108 108 Chapter 2 Geometric Reasoning (1, y)NorthpointOverlook Northcampground(3, 5)(x, 1)WaterfallFinal file 2/25/05Campground mapHolt Rinehart WinstonGeometry 2007 Texasge07sec02l05003a Karen Minot(415)883-6560�� 37. This problem will prepare you for the Multi-Step TAKS Prep on page 126. Recall from Algebra 1 that the Multiplication and Division Properties of Inequality tell you to reverse the inequality sign when multiplying or dividing by a negative number. a. Solve the inequality x + 15 ≤ 63 and write a justification for each step. b. Solve the inequality -2x > 36 and write a justification for each step. 38. Write About It Compare the conclusion of a deductive proof and a conjecture based on inductive reasoning. 39. Which could NOT be used to justify the statement ̶̶ AB ≅ ̶̶ CD? Definition of congruence Symmetric Property of Congruence Reflexive Property of Congruence Transitive Property of Congruence 40. A club membership costs $35 plus $3 each time t the member uses the pool. Which equation represents the total cost C of the membership? C + 35 = 3t C = 35 + 3t 35 = C + 3t 41. Which statement is true by the Reflexive Property of Equality? ̶̶ CD = ̶̶ RT ≅ x = 35 ̶̶ CD ̶̶ TR C = 35t + 3 CD = CD 42. Gridded Response In the triangle, m∠1 + m∠2 + m∠3 = 180°. If m∠3 = 2m∠1 and m∠1 = m∠2, find m∠3 in degrees. CHALLENGE AND EXTEND 43. In the gate, PA = QB, QB = RA, and PA = 18 in. Find PR, and justify each step. 44. Critical Thinking Explain why there is no Addition Property of Congruence. 45. Algebra Justify each step in the solution of the inequality 7 - 3x > 19. SPIRAL REVIEW 46. The members of a high

school band have saved $600 for a trip. They deposit the money in a savings account. What additional information is needed to find the amount of interest the account earns during a 3-month period? (Previous course) Use a compass and straightedge to construct each of the following. (Lesson 1-2) 47. ̶̶ JK congruent to ̶̶̶ MN 48. a segment bisector of ̶̶ JK Identify whether each conclusion uses inductive or deductive reasoning. (Lesson 2-3) 49. A triangle is obtuse if one of its angles is obtuse. Jacob draws a triangle with two acute angles and one obtuse angle. He concludes that the triangle is obtuse. 50. Tonya studied 3 hours for each of her last two geometry tests. She got an A on both tests. She concludes that she will get an A on the next test if she studies for 3 hours. 2- 5 Algebraic Proof 109 109 �� 2-6 Geometric Proof TEKS G.1.A Geometric structure: develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems. Also G.3.B, G.3.C, G.3.E Objectives Write two-column proofs. Prove geometric theorems by using deductive reasoning. Vocabulary theorem two-column proof Who uses this? To persuade your parents to increase your allowance, your argument must be presented logically and precisely. When writing a geometric proof, you use deductive reasoning to create a chain of logical steps that move from the hypothesis to the conclusion of the conjecture you are proving. By proving that the conclusion is true, you have proven that the original conjecture is true When writing a proof, it is important to justify each logical step with a reason. You can use symbols and abbreviations, but they must be clear enough so that anyone who reads your proof will understand them. E X A M P L E 1 Writing Justifications When a justification is based on more than the previous step, you can note this after the reason, as in Example 1 Step 5. Write a justification for each step, given that ∠ A and ∠B are complementary and ∠ A ≅ ∠C. 1. ∠ A and ∠B are complementary. 2. m∠ A + m∠B = 90° 3. ∠

A ≅ ∠C 4. m∠ A = m∠C 5. m∠C + m∠B = 90° 6. ∠C and ∠B are complementary. Given information Def. of comp.  Given information Def. of ≅  Subst. Prop. of = Def. of comp.  Steps 2, 4 1. Write a justification for each step, given that B is the midpoint ̶̶ AC and ̶̶ AB ≅ ̶̶ EF. of 1. B is the midpoint of 2. 3. 4. ̶̶ AB ≅ ̶̶ AB ≅ ̶̶ BC ≅ ̶̶ BC ̶̶ EF ̶̶ EF ̶̶ AC. A theorem is any statement that you can prove. Once you have proven a theorem, you can use it as a reason in later proofs. Theorem THEOREM HYPOTHESIS CONCLUSION 2-6-1 Linear Pair Theorem If two angles form a linear pair, then they are supplementary. ∠A and ∠B form a linear pair. ∠A and ∠B are supplementary. 110 110 Chapter 2 Geometric Reasoning �� Theorem THEOREM HYPOTHESIS CONCLUSION 2-6-2 Congruent Supplements Theorem If two angles are supplementary to the same angle (or to two congruent angles), then the two angles are congruent. ∠1 and ∠2 are supplementary. ∠2 and ∠3 are supplementary. ∠1 ≅ ∠3 A geometric proof begins with Given and Prove statements, which restate the hypothesis and conclusion of the conjecture. In a two-column proof, you list the steps of the proof in the left column. You write the matching reason for each step in the right column. E X A M P L E 2 Completing a Two-Column Proof Fill in the blanks to complete a two-column proof of the Linear Pair Theorem. Given: ∠1 and ∠2 form a linear pair. Prove: ∠1 and ∠2 are supplementary. Proof: Statements Reasons 1. ∠1 and ∠2 form a linear pair. 1. Given 2.  BA and �

� BC form a line. 3. m∠ABC = 180° 4. a. 5. b.? ̶̶̶̶̶̶? ̶̶̶̶̶̶ 6. ∠1 and ∠2 are supplementary. 2. Def. of lin. pair 3. Def. of straight ∠ 4. ∠ Add. Post. 5. Subst. Steps 3, 4 6. c.? ̶̶̶̶̶̶ Since there is no other substitution property, the Substitution Property of Equality is often written as “Substitution” or “Subst.” Use the existing statements and reasons in the proof to fill in the blanks. a. m∠1 + m∠2 = m∠ABC b. m∠1 + m∠2 = 180° c. Def. of supp.  The ∠ Add. Post. is given as the reason. Substitute 180° for m∠ABC. The measures of supp.  add to 180° by def. 2. Fill in the blanks to complete a two-column proof of one case of the Congruent Supplements Theorem. Given: ∠1 and ∠2 are supplementary, and ∠2 and ∠3 are supplementary. Prove: ∠1 ≅ ∠3 Proof: Statements Reasons 1. a.? ̶̶̶̶̶̶ 2. m∠1 + m∠2 = 180° m∠2 + m∠3 = 180° 3. b.? ̶̶̶̶̶̶ 4. m∠2 = m∠2 5. m∠1 = m∠3 6. d.? ̶̶̶̶̶̶ 1. Given 2. Def. of supp.  3. Subst. 4. Reflex. Prop. of = 5. c.? ̶̶̶̶̶̶ 6. Def. of ≅  2- 6 Geometric Proof 111 111 �� Before you start writing a proof, you should plan out your logic. Sometimes you will be given a plan for a more challenging proof. This plan will detail the major steps of the proof for you. Theorems THEOREM HYPOTHESIS CONCLUSION 2-6-3 Right Angle Congruence Theorem All right angles are congruent. 2-6-4 Congruent Comple

ments Theorem If two angles are complementary to the same angle (or to two congruent angles), then the two angles are congruent. ∠A and ∠B are right angles. ∠A ≅ ∠B ∠1 and ∠2 are complementary. ∠2 and ∠3 are complementary. ∠1 ≅ ∠3 E X A M P L E 3 Writing a Two-Column Proof from a Plan If a diagram for a proof is not provided, draw your own and mark the given information on it. But do not mark the information in the Prove statement on it. Use the given plan to write a two-column proof of the Right Angle Congruence Theorem. Given: ∠1 and ∠2 are right angles. Prove: ∠1 ≅ ∠2 Plan: Use the definition of a right angle to write the measure of each angle. Then use the Transitive Property and the definition of congruent angles. Proof: Statements Reasons 1. ∠1 and ∠2 are right angles. 1. Given 2. m∠1 = 90°, m∠2 = 90° 3. m∠1 = m∠2 4. ∠1 ≅ ∠2 2. Def. of rt. ∠ 3. Trans. Prop. of = 4. Def. of ≅  3. Use the given plan to write a two-column proof of one case of the Congruent Complements Theorem. Given: ∠1 and ∠2 are complementary, and ∠2 and ∠3 are complementary. Prove: ∠1 ≅ ∠3 Plan: The measures of complementary angles add to 90° by definition. Use substitution to show that the sums of both pairs are equal. Use the Subtraction Property and the definition of congruent angles to conclude that ∠1 ≅ ∠3. The Proof Process 1. Write the conjecture to be proven. 2. Draw a diagram to represent the hypothesis of the conjecture. 3. State the given information and mark it on the diagram. 4. State the conclusion of the conjecture in terms of the diagram. 5. Plan your argument and prove the conjecture. 112 112 Chapter 2 Geometric Reasoning �� THINK AND DISCUSS 1. Which step in a proof should match the Prove statement? 2. Why is it important to include every logical step in

a proof? 3. List four things you can use to justify a step in a proof. 4. GET ORGANIZED Copy and complete the graphic organizer. In each box, describe the steps of the proof process. 2-6 Exercises Exercises KEYWORD: MG7 2-6 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. In a two-column proof, you list the? in the left column and the ̶̶̶̶ the right column. (statements or reasons)? in ̶̶̶̶. Write a justification for each step, given that m∠A = 60° and m∠B = 2m∠A. 2. A? is a statement you can prove. (postulate or theorem) ̶̶̶̶ p. 110 1. m∠A = 60°, m∠B = 2m∠A 2. m∠B = 2 (60°) 3. m∠B = 120° 4. m∠A + m∠ B = 60° + 120° 5. m∠A + m∠B = 180° 6. ∠A and ∠B are supplementary. Fill in the blanks to complete the two-column proof. p. 111 Given: ∠2 ≅ ∠3 Prove: ∠1 and ∠3 are supplementary. Proof: Statements Reasons 1. ∠2 ≅ ∠3 2. m∠2 = m∠3 3. b.? ̶̶̶̶̶̶ 1. Given 2. a.? ̶̶̶̶̶̶ 3. Lin. Pair Thm. 4. m∠1 + m∠2 = 180° 4. Def. of supp.  5. m∠1 + m∠3 = 180° 6. d.? ̶̶̶̶̶̶ 5. c.? Steps 2, 4 ̶̶̶̶̶ 6. Def. of supp. 112 5. Use the given plan to write a two-column proof. Given: X is the midpoint of ̶̶ YB Prove: ̶̶ AX ≅ ̶̶ AY, and Y is the midpoint of ̶̶ XB. Plan: By the definition of midpoint, Use the Transitive Property to conclude

that ̶̶ AX ≅ ̶̶ XY, and ̶̶ AX ≅ ̶̶ XY ≅ ̶̶ YB. ̶̶ YB. 2- 6 Geometric Proof 113 113 �� Independent Practice For See Exercises Example 6 7–8 9–10 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S7 Application Practice p. S29 PRACTICE AND PROBLEM SOLVING 6. Write a justification for each step, given that  BX bisects ∠ABC and m∠XBC = 45°. 1.  BX bisects ∠ABC. 2. ∠ABX ≅ ∠XBC 3. m∠ABX = m∠XBC 4. m∠XBC = 45° 5. m∠ABX = 45° 6. m∠ABX + m∠XBC = m∠ABC 7. 45° + 45° = m∠ABC 8. 90° = m∠ABC 9. ∠ABC is a right angle. Fill in the blanks to complete each two-column proof. 7. Given: ∠1 and∠2 are supplementary, and ∠3 and ∠4 are supplementary. ∠2 ≅ ∠3 Prove: ∠1 ≅ ∠4 Proof: Statements Reasons 1. ∠1 and ∠2 are supplementary. ∠3 and ∠4 are supplementary. 1. Given 2. a.? ̶̶̶̶̶ 3. m∠1 + m∠2 = m∠3 + m∠4 4. ∠2 ≅ ∠3 5. m∠2 = m∠3 6. c.? ̶̶̶̶̶ 7. ∠1 ≅ ∠4 2. Def. of supp.  3. b.? ̶̶̶̶̶ 4. Given 5. Def. of ≅  6. Subtr. Prop. of = Steps 3, 5 7. d.? ̶̶̶̶̶ 8. Given: ∠BAC is a right angle. ∠2 ≅ ∠3 Prove: ∠1 and ∠3 are complementary. Proof: Statements Reasons 1. �

�BAC is a right angle. 1. Given 2. m∠BAC = 90° 3. b.? ̶̶̶̶̶ 2. a.? ̶̶̶̶̶ 3. ∠ Add. Post. 4. m∠1 + m∠2 = 90° 4. Subst. Steps 2, 3 5. ∠2 ≅ ∠3 6. c.? ̶̶̶̶̶ 7. m∠1 + m∠3 = 90° 8. e.? ̶̶̶̶̶ 5. Given 6. Def. of ≅  7. d.? Steps 4, 6 ̶̶̶̶̶ 8. Def. of comp.  Use the given plan to write a two-column proof. 9. Given: Prove: ̶̶ BE ≅ ̶̶ AB ≅ ̶̶ CE, ̶̶ CD ̶̶ DE ≅ ̶̶ AE Plan: Use the definition of congruent segments to write the given information in terms of lengths. Then use the Segment Addition Postulate to show that AB = CD and thus ̶̶ AB ≅ ̶̶ CD. 114 114 Chapter 2 Geometric Reasoning �� Use the given plan to write a two-column proof. 10. Given: ∠1 and ∠3 are complementary, and ∠2 and ∠4 are complementary. ∠3 ≅ ∠4 Prove: ∠1 ≅ ∠2 Plan: Since ∠1 and ∠3 are complementary and ∠2 and ∠4 are complementary, both pairs of angle measures add to 90°. Use substitution to show that the sums of both pairs are equal. Since ∠3 ≅ ∠4, their measures are equal. Use the Subtraction Property of Equality and the definition of congruent angles to conclude that ∠1 ≅ ∠2. Engineering Find each angle measure. Find each angle measure. 11. 11. m∠1 12. m∠2 13. m∠3 The Bluff Dale Bridge, constructed by Texas builder William Flinn in 1891, is the oldest known cable-stayed bridge in the United States. 14. Engineering The Bluff Dale Bridge is 140 feet long and spans the Paluxy River in Bluff Dale, Texas. If ∠1 ≅ ∠2, which

theorem can you use to conclude that ∠3 ≅ ∠4? 15. Critical Thinking Explain why there are two cases to consider when proving the Congruent Supplements Theorem and the Congruent Complements Theorem. Tell whether each statement is sometimes, always, or never true. 16. An angle and its complement are congruent. 17. A pair of right angles forms a linear pair. 18. An angle and its complement form a right angle. 19. A linear pair of angles is complementary. Algebra Find the value of each variable. 20. 21. 22. � � � � 23. Write About It How are a theorem and a postulate alike? How are they different? 24. This problem will prepare you for the Multi-Step TAKS Prep on page 126. Sometimes you may be asked to write a proof without a specific statement of the Given and Prove information being provided for you. For each of the following situations, use the triangle to write a Given and Prove statement. a. The segment connecting the midpoints of two sides of a triangle is half as long as the third side. b. The acute angles of a right triangle are complementary. c. In a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse. 2- 6 Geometric Proof 115 115 �� 25. Which theorem justifies the conclusion that ∠1 ≅ ∠4? Linear Pair Theorem Congruent Supplements Theorem Congruent Complements Theorem Right Angle Congruence Theorem 26. What can be concluded from the statement m∠1 + m∠2 = 180°? ∠1 and ∠2 are congruent. ∠1 and ∠2 are supplementary. ∠1 and ∠2 are complementary. ∠1 and ∠2 form a linear pair. 27. Given: Two angles are complementary. The measure of one angle is 10° less than the measure of the other angle. Conclusion: The measures of the angles are 85° and 95°. Which statement is true? The conclusion is correct because 85° is 10° less than 95°. The conclusion is verified by the first statement given. The conclusion is invalid because the angles are not congruent. The conclusion is contradicted by the first statement given. CHALLENGE AND EXTEND 28. Write a

two-column proof. Given: m∠LAN = 30°, m∠1 = 15° Prove:  AM bisects ∠LAN. Multi-Step Find the value of the variable and the measure of each angle. 29. 30. SPIRAL REVIEW The table shows the number of tires replaced by a repair company during one week, classified by the mileage on the tires when they were replaced. Use the table for Exercises 31 and 32. (Previous course) 31. What percent of the tires had mileage between 40,000 and 49,999 when replaced? 32. If the company replaces twice as many tires next week, about how many tires would you expect to have lasted between 80,000 and 89,999 miles? Mileage on Replaced Tires Mileage Tires 40,000–49,999 50,000–59,999 60,000–69,999 70,000–79,999 80,000–89,999 60 82 54 40 14 Sketch a figure that shows each of the following. (Lesson 1-1) 33. Through any two collinear points, there is more than one plane containing them. 34. A pair of opposite rays forms a line. Identify the property that justifies each statement. (Lesson 2-5) 35. ̶̶ JK ≅ ̶̶ KL, so ̶̶ KL ≅ ̶̶ JK. 36. If m = n and n = p, then m = p. 116 116 Chapter 2 Geometric Reasoning �� 2-6 Use with Lesson 2-6 Activity Design Plans for Proofs Sometimes the most challenging part of writing a proof is planning the logical steps that will take you from the Given statement to the Prove statement. Like working a jigsaw puzzle, you can start with any piece. Write down everything you know from the Given statement. If you don’t see the connection right away, start with the Prove statement and work backward. Then connect the pieces into a logical order. TEKS G.1.A Geometric structure: develop an awareness of the structure of a mathematical system... Also G.3.B, G.3.C, G.3.E Prove the Common Angles Theorem. Given: ∠AXB ≅ ∠CXD Prove: ∠AXC

≅ ∠BXD 1 Start by considering the difference in the Given and Prove statements. How does ∠AXB compare to ∠AXC? How does ∠CXD compare to ∠BXD? In both cases, ∠BXC is combined with the first angle to get the second angle. 2 The situation involves combining adjacent angle measures, so list any definitions, properties, postulates, and theorems that might be helpful. Definition of congruent angles, Angle Addition Postulate, properties of equality, and Reflexive, Symmetric, and Transitive Properties of Congruence 3 Start with what you are given and what you are trying to prove and then work toward the middle. ∠AXB ≅ ∠CXD m∠AXB = m∠CXD??? m∠AXC = m∠BXD ∠AXC ≅ ∠BXD The first reason will be “Given.” Def. of ≅ ??? Def. of ≅  The last statement will be the Prove statement. 4 Based on Step 1, ∠BXC is the missing piece in the middle of the logical flow. So write down what you know about ∠BXC. ∠BXC ≅ ∠BXC m∠BXC = m∠BXC Reflex. Prop. of ≅ Reflex. Prop. of = 5 Now you can see that the Angle Addition Postulate needs to be used to complete the proof. m∠AXB + m∠BXC = m∠AXC m∠BXC + m∠CXD = m∠BXD ∠ Add. Post. ∠ Add. Post. 6 Reorder the pieces above to write a two-column proof of the Common Angles Theorem. Try This 1. Describe how a plan for a proof differs from the actual proof. 2. Write a plan and a two-column proof. BD bisects ∠ABC. Given: Prove: 2m∠1 = m∠ABC 3. Write a plan and a two-column proof. Given: ∠LXN is a right angle. Prove: ∠1 and ∠2 are complementary. 2-6 Geometry Lab 117 117 ��

�� 2-7 Flowchart and Paragraph Proofs TEKS G.1.A Geometric structure: develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems. Also G.2.B, G.3.C, G.3.E Objectives Write flowchart and paragraph proofs. Prove geometric theorems by using deductive reasoning. Vocabulary flowchart proof paragraph proof Why learn this? Flowcharts make it easy to see how the steps of a process are linked together. A second style of proof is a flowchart proof, which uses boxes and arrows to show the structure of the proof. The steps in a flowchart proof move from left to right or from top to bottom, shown by the arrows connecting each box. The justification for each step is written below the box. Theorem 2-7-1 Common Segments Theorem THEOREM HYPOTHESIS CONCLUSION Given collinear points A, B, C, and D arranged ̶̶ CD, then as shown, if ̶̶ AC ≅ ̶̶ AB ≅ ̶̶ BD. ̶̶ AB ≅ ̶̶ CD ̶̶ AC ≅ ̶̶ BD E X A M P L E 1 Reading a Flowchart Proof Use the given flowchart proof to write a two-column proof of the Common Segments Theorem. Given: Prove: ̶̶ AB ≅ ̶̶ AC ≅ ̶̶ CD ̶̶ BD Flowchart proof: �� Two-column proof: Statements Reasons ̶̶ AB ≅ ̶̶ CD 1. 2. AB = CD 3. BC = BC 4. AB + BC = BC + CD 1. Given 2. Def. of ≅ segs. 3. Reflex. Prop. of = 4. Add. Prop. of = 5. AB + BC = AC, BC + CD = BD 5. Seg. Add. Post. 6. AC = BD ̶̶ BD ̶̶ AC ≅ 7. 6. Subst. 7. Def. of ≅ segs. 118 118 Chapter 2 Geometric Reasoning �� 1. Use the given flowchart

proof to write a two-column proof. Given: RS = UV, ST = TU Prove: ̶̶ RT ≅ ̶̶ TV Flowchart proof: E X A M P L E 2 Writing a Flowchart Proof Use the given two-column proof to write a flowchart proof of the Converse of the Common Segments Theorem. Given: Prove: ̶̶ AC ≅ ̶̶ AB ≅ ̶̶ BD ̶̶ CD Two-column proof: Statements Reasons Like the converse of a conditional statement, the converse of a theorem is found by switching the hypothesis and conclusion. ̶̶ AC ≅ ̶̶ BD 1. 2. AC = BD 1. Given 2. Def. of ≅ segs. 3. AB + BC = AC, BC + CD = BD 3. Seg. Add. Post. 4. AB + BC = BC + CD 5. BC = BC 6. AB = CD ̶̶ CD ̶̶ AB ≅ 7. 4. Subst. Steps 2, 3 5. Reflex. Prop. of = 6. Subtr. Prop. of = 7. Def. of ≅ segs. Flowchart proof: 2. Use the given two-column proof to write a flowchart proof. Given: ∠2 ≅ ∠4 Prove: m∠1 = m∠3 Two-column proof: Statements Reasons 1. ∠2 ≅ ∠4 2. ∠1 and ∠2 are supplementary. ∠3 and ∠4 are supplementary. 3. ∠1 ≅ ∠3 4. m∠1 = m∠3 1. Given 2. Lin. Pair Thm. 3. ≅ Supps. Thm. 4. Def. of ≅  2-7 Flowchart and Paragraph Proofs 119 119 �� A paragraph proof is a style of proof that presents the steps of the proof and their matching reasons as sentences in a paragraph. Although this style of proof is less formal than a two-column proof, you still

must include every step. Theorems THEOREM HYPOTHESIS CONCLUSION 2-7-2 Vertical Angles Theorem Vertical angles are congruent. ∠A and ∠B are vertical angles. ∠A ≅ ∠B 2-7-3 If two congruent angles are supplementary, then each angle is a right angle. (≅  supp. → rt. ) ∠1 ≅ ∠2 ∠1 and ∠2 are supplementary. ∠1 and ∠2 are right angles. E X A M P L E 3 Reading a Paragraph Proof Use the given paragraph proof to write a two-column proof of the Vertical Angles Theorem. Given: ∠1 and ∠3 are vertical angles. Prove: ∠1 ≅ ∠3 � � � Paragraph proof: ∠1 and ∠3 are vertical angles, so they are formed by intersecting lines. Therefore ∠1 and ∠2 are a linear pair, and ∠2 and ∠3 are a linear pair. By the Linear Pair Theorem, ∠1 and ∠2 �� are supplementary, and ∠2 and ∠3 are supplementary. So by the �� Congruent Supplements Theorem, ∠1 ≅ ∠3. �� Two-column proof: Statements Reasons 1. ∠1 and ∠3 are vertical angles. 1. Given 2. ∠1 and ∠3 are formed by intersecting lines. 2. Def. of vert.  3. ∠1 and ∠2 are a linear pair. ∠2 and ∠3 are a linear pair. 4. ∠1 and ∠2 are supplementary. ∠2 and ∠3 are supplementary. 5. ∠1 ≅ ∠3 3. Def. of lin. pair 4. Lin. Pair Thm. 5. ≅ Supps. Thm. 3. Use the given paragraph proof to write a two-column proof. Given: ∠WXY is a right angle. ∠1 ≅ ∠3 Prove: ∠1 and ∠2 are complementary. Paragraph proof: Since ∠WXY is a right angle, m∠WXY = 90° by the definition of a right angle. By the Angle Addition Postulate, m∠WXY

= m∠2 + m∠3. By substitution, m∠2 + m∠3 = 90°. Since ∠1 ≅ ∠3, m∠1 = m∠3 by the definition of congruent angles. Using substitution, m∠2 + m∠1 = 90°. Thus by the definition of complementary angles, ∠1 and ∠2 are complementary. 120 120 Chapter 2 Geometric Reasoning �� Writing a Proof When I have to write a proof and I don’t see how to start, I look at what I’m supposed to be proving and see if it makes sense. If it does, I ask myself why. Sometimes this helps me to see what the reasons in the proof might be. If all else fails, I just start writing down everything I know based on the diagram and the given statement. By brainstorming like this, I can usually figure out the steps of the proof. You can even write each thing on a separate piece of paper and arrange the pieces of paper like a flowchart. Claire Jeffords Riverbend High School E X A M P L E 4 Writing a Paragraph Proof Use the given two-column proof to write a paragraph proof of Theorem 2-7-3. Given: ∠1 and ∠2 are supplementary. ∠1 ≅ ∠2 Prove: ∠1 and ∠2 are right angles. Two-column proof: Statements Reasons 1. ∠1 and ∠2 are supplementary. 1. Given ∠1 ≅ ∠2 2. m∠1 + m∠2 = 180° 3. m∠1 = m∠2 4. m∠1 + m∠1 = 180° 5. 2m∠1 = 180° 6. m∠1 = 90° 7. m∠2 = 90° 2. Def. of supp.  3. Def. of ≅  Step 1 4. Subst. Steps 2, 3 5. Simplification 6. Div. Prop. of = 7. Trans. Prop. of = Steps 3, 6 8. ∠1 and ∠2 are right angles. 8. Def. of rt. ∠ Paragraph proof: ∠1 and ∠2 are supplementary, so m∠1 + m∠2 = 180° by the definition of supplementary angles. They are

also congruent, so their measures are equal by the definition of congruent angles. By substitution, m∠1 + m∠1 = 180°, so m∠1 = 90° by the Division Property of Equality. Because m∠1 = m∠2, m∠2 = 90° by the Transitive Property of Equality. So both are right angles by the definition of a right angle. 4. Use the given two-column proof to write a paragraph proof. Given: ∠1 ≅ ∠4 Prove: ∠2 ≅ ∠3 Two-column proof: Statements Reasons 1. ∠1 ≅ ∠4 1. Given 2. ∠1 ≅ ∠2, ∠3 ≅ ∠4 2. Vert.  Thm. 3. ∠2 ≅ ∠4 4. ∠2 ≅ ∠3 3. Trans. Prop. of ≅ Steps 1, 2 4. Trans. Prop. of ≅ Steps 2, 3 2-7 Flowchart and Paragraph Proofs 121 121 �� THINK AND DISCUSS 1. Explain why there might be more than one correct way to write a proof. 2. Describe the steps you take when writing a proof. 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, describe the proof style in your own words. 2-7 Exercises Exercises KEYWORD: MG7 2-7 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. In a? proof, the logical order is represented by arrows that connect each step. ̶̶̶̶ (flowchart or paragraph) 2. The steps and reasons of a (flowchart or paragraph)? proof are written out in sentences. ̶̶̶̶. Use the given flowchart proof to write p. 118 a two-column proof. Given: ∠1 ≅ ∠2 Prove: ∠1 and ∠2 are right angles. Flowchart proof. Use the given two-column proof to write p. 119 a flowchart proof. Given: ∠2 and ∠4 are supplementary. Prove: m∠2 = m∠3 Two-column proof: Statements Reasons 1. ∠2 and ∠4 are supplementary. 1. Given 2. �

�3 and ∠4 are supplementary. 2. Lin. Pair Thm. 3. ∠2 ≅ ∠3 4. m∠2 = m∠3 3. ≅ Supps. Thm. Steps 1, 2 4. Def. of ≅  122 122 Chapter 2 Geometric Reasoning ��. Use the given paragraph proof to write a two-column proof. p. 120 Given: ∠2 ≅ ∠4 Prove: ∠1 ≅ ∠3 Paragraph proof: By the Vertical Angles Theorem, ∠1 ≅ ∠2, and ∠3 ≅ ∠4. It is given that ∠2 ≅ ∠4. By the Transitive Property of Congruence, ∠1 ≅ ∠4, and thus ∠1 ≅ ∠3. Use the given two-column proof to write a paragraph proof. p. 121 Given: Prove:  BD bisects ∠ABC.  BG bisects ∠FBH. Two-column proof: Statements Reasons 1.  BD bisects ∠ABC. 1. Given 2. ∠1 ≅ ∠2 2. Def. of ∠ bisector 3. ∠1 ≅ ∠4, ∠2 ≅ ∠3 3. Vert.  Thm. 4. ∠4 ≅ ∠2 5. ∠4 ≅ ∠3 4. Trans. Prop. of ≅ Steps 2, 3 5. Trans. Prop. of ≅ Steps 3, 4 6.  BG bisects ∠FBH. 6. Def. of ∠ bisector Independent Practice For See Exercises Example 7 8 9 10 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S7 Application Practice p. S29 PRACTICE AND PROBLEM SOLVING 7. Use the given flowchart proof to write a two-column proof. Given: B is the midpoint of ̶̶ AC. AD = EC Prove: DB

= BE Flowchart proof: 8. Use the given two-column proof to write a flowchart proof. Given: ∠3 is a right angle. Prove: ∠4 is a right angle. Two-column proof: Statements Reasons 1. ∠3 is a right angle. 2. m∠3 = 90° 1. Given 2. Def. of rt. ∠ 3. ∠3 and ∠4 are supplementary. 3. Lin. Pair Thm. 4. m∠3 + m∠4 = 180° 5. 90° + m∠4 = 180° 6. m∠4 = 90° 7. ∠4 is a right angle. 4. Def. of supp.  5. Subst. Steps 2, 4 6. Subtr. Prop. of = 7. Def. of rt. ∠ 2-7 Flowchart and Paragraph Proofs 123 123 �� 9. Use the given paragraph proof to write a two-column proof. Given: ∠1 ≅ ∠4 Prove: ∠2 and ∠3 are supplementary. Paragraph proof: ∠4 and ∠3 form a linear pair, so they are supplementary by the Linear Pair Theorem. Therefore, m∠4 + m∠3 = 180°. Also, ∠1 and ∠2 are vertical angles, so ∠1 ≅ ∠2 by the Vertical Angles Theorem. It is given that ∠1 ≅ ∠4. So by the Transitive Property of Congruence, ∠4 ≅ ∠2, and by the definition of congruent angles, m∠4 = m∠2. By substitution, m∠2 + m∠3 = 180°, so ∠2 and ∠3 are supplementary by the definition of supplementary angles. 10. Use the given two-column proof to write a paragraph proof. Given: ∠1 and ∠2 are complementary. Prove: ∠2 and ∠3 are complementary. Two-column proof: Statements Reasons 1. ∠1 and ∠2 are complementary. 1. Given 2. m∠1 + m∠

2 = 90° 3. ∠1 ≅ ∠3 4. m∠1 = m∠3 5. m∠3 + m∠2 = 90° 2. Def. of comp.  3. Vert.  Thm. 4. Def. of ≅  5. Subst. Steps 2, 4 6. ∠2 and ∠3 are complementary. 6. Def. of comp.  Find each measure and name the theorem that justifies your answer. 11. AB 12. m∠2 13. m∠3 Algebra Find the value of each variable. 14. 15. 16. 17. /////ERROR ANALYSIS///// Below are two drawings for the given proof. Which is incorrect? Explain the error. ̶̶ BC Given: Prove: ∠A ≅ ∠C ̶̶ AB ≅ 18. This problem will prepare you for the Multi-Step TAKS Prep on page 126. Rearrange the pieces to create a flowchart proof. 124 124 Chapter 2 Geometric Reasoning ��x � ��x � ��y�� 19. Critical Thinking Two lines intersect, and one of the angles formed is a right angle. Explain why all four angles are congruent. 20. Write About It Which style of proof do you find easiest to write? to read? 21. Which pair of angles in the diagram must be congruent? ∠1 and ∠5 ∠3 and ∠4 ∠5 and ∠8 None of the above 22. What is the measure of ∠2? 38° 52° 128° 142° 23. Which statement is NOT true if ∠2 and ∠6 are supplementary? m∠2 + m∠6 = 180° ∠2 and ∠3 are supplementary. ∠1 and ∠6 are supplementary. m∠1 + m∠4 = 180° CHALLENGE AND EXTEND 24. Textiles Use the woven pattern to write a flowchart proof. Given: ∠1 ≅ ∠3 Prove: m∠4 + m∠5 = m�

�6 25. Write a two-column proof. Given: ∠AOC ≅ ∠BOD Prove: ∠AOB ≅ ∠COD 26. Write a paragraph proof. Given: ∠2 and ∠5 are right angles. m∠1 + m∠2 + m∠3 = m∠4 + m∠5 + m∠6 Prove: ∠1 ≅ ∠4 27. Multi-Step Find the value of each variable and the measures of all four angles. SPIRAL REVIEW Solve each system of equations. Check your solution. (Previous course) ⎧ 7x - y = -33 29. ⎨ 3x + y = -7 ⎩ ⎧ 28. ⎨ ⎩ y = -6x + 18 y = 2x + 14 ⎧ ⎨ ⎩ 30. 2x + y = 8 -x + 3y = 10 Use a protractor to draw an angle with each of the following measures. (Lesson 1-3) 31. 125° 32. 38° 33. 94° 34. 175° For each conditional, write the converse and a biconditional statement. (Lesson 2-4) 35. If a positive integer has more than two factors, then it is a composite number. 36. If a quadrilateral is a trapezoid, then it has exactly one pair of parallel sides. 2-7 Flowchart and Paragraph Proofs 125 125 �� SECTION 2B Mathematical Proof Intersection Inspection According to the U.S. Department of Transportation, it is ideal for two intersecting streets to form four 90° angles. If this is not possible, roadways should meet at an angle of 75° or greater for maximum safety and visibility. 1. Write a compound inequality to represent the range of measures an angle in an intersection should have. 2. Suppose that an angle in an intersection meets the guidelines specified by the U.S. Department of Transportation. Find the range of measures for the adjacent angle in the intersection. The intersection of West Elm Street and Lamar Boulevard has a history of car accidents. The Southland neighborhood association is circulating a petition to have the city reconstruct the intersection. A surveyor measured the intersection, and one of the angles measures

145°. 3. Given that m∠2 = 145°, write a two-column proof to show that m∠1 and m∠3 are less than 75°. 4. Write a paragraph proof to justify the argument that the intersection of West Elm Street and Lamar Boulevard should be reconstructed. 126 126 Chapter 2 Geometric Reasoning �� SECTION 2B Quiz for Lessons 2-5 Through 2-7 2-5 Algebraic Proof Solve each equation. Write a justification for each step. 1. m - 8 = 13 2. 4y - 1 = 27 3. - x _ 3 = 2 Identify the property that justifies each statement. 4. m∠XYZ = m∠PQR, so m∠PQR = m∠XYZ. ̶̶ AB ≅ ̶̶ AB 5. 6. ∠4 ≅ ∠A, and ∠A ≅ ∠1. So ∠4 ≅ ∠1. 7. k = 7, and m = 7. So k = m. 2-6 Geometric Proof 8. Fill in the blanks to complete the two-column proof. Given: m∠1 + m∠3 = 180° Prove: ∠1 ≅ ∠4 Proof: Statements Reasons 1. m∠1 + m∠3 = 180° 2. b.? ̶̶̶̶̶̶ 1. a.? ̶̶̶̶̶̶ 2. Def. of supp.  3. ∠3 and ∠4 are supplementary. 3. Lin. Pair Thm. 4. ∠3 ≅ ∠3 5. d.? ̶̶̶̶̶̶ 4. c.? ̶̶̶̶̶ 5. ≅ Supps. Thm. 9. Use the given plan to write a two-column proof of the Symmetric Property of ̶̶ EF ̶̶ AB Congruence. ̶̶ AB ≅ Given: ̶̶ EF ≅ Prove: Plan: Use the definition of congruent segments to write of equality. Then use the Symmetric Property of Equality to show that EF = AB. So ̶̶ AB by the definition of congruent segments. ̶̶ EF as a statement ̶̶ AB ≅ ̶̶ EF ≅ 2-7 Flowchart and Par

agraph Proofs Use the given two-column proof to write the following. Given: ∠1 ≅ ∠3 Prove: ∠2 ≅ ∠4 Proof: Statements Reasons 1. ∠1 ≅ ∠3 1. Given 2. ∠1 ≅ ∠2, ∠3 ≅ ∠4 2. Vert.  Thm. 3. ∠2 ≅ ∠3 4. ∠2 ≅ ∠4 3. Trans. Prop. of ≅ 4. Trans. Prop. of ≅ 10. a flowchart proof 11. a paragraph proof Ready to Go On? 127 127 �� EXTENSION EXTENSION Introduction to Symbolic Logic Objectives Analyze the truth value of conjunctions and disjunctions. Construct truth tables to determine the truth value of logical statements. Vocabulary compound statement conjunction disjunction truth table TEKS G.4.A Select an appropriate representation... in order to solve problems. Also G.3.C Symbolic logic is used by computer programmers, mathematicians, and philosophers to analyze the truth value of statements, independent of their actual meaning. A compound statement is created by combining two or more statements. Suppose p and q each represent a statement. Two compound statements can be formed by combining p and q: a conjunction and a disjunction. Compound Statements TERM WORDS SYMBOLS EXAMPLE Conjunction A compound statement that uses the word and p AND q p ⋀ q Pat is a band member AND Pat plays tennis. Disjunction A compound statement that uses the word or p OR q p ⋁ q Pat is a band member OR Pat plays tennis. A conjunction is true only when all of its parts are true. A disjunction is true if any one of its parts is true. E X A M P L E 1 Analyzing Truth Values of Conjunctions and Disjunctions Use p, q, and r to find the truth value of each compound statement. p: Washington, D.C., is the capital of the United States. q: The day after Monday is Tuesday. r: California is the largest state in the United States. A q ⋁ r B r ⋀ p Since q is true, the disjunction is true. Since r is false, the conjunction is false. Use the information given above to find the truth value of each compound statement. 1a. r ⋁ p

1b. p ⋀ q A table that lists all possible combinations of truth values for a statement is called a truth table. A truth table shows you the truth value of a compound statement, based on the possible truth values of its parts. Make sure you include all possible combinations of truth values for each piece of the compound statement 128 128 Chapter 2 Geometric Reasoning E X A M P L E 2 Constructing Truth Tables for Compound Statements Construct a truth table for the compound statement ∼u ⋀ (v ⋁ w). Since u, v, and w can each be either true or false, the truth table will have (2) (2) (2) = 8 rows. The negation (~) of a statement has the opposite truth valueu v ⋁ w ∼u ⋀ (v ⋁ w. Construct a truth table for the compound statement ∼u ⋀ ∼v. EXTENSION Exercises Exercises Use p, q, and r to find the truth value of each compound statement. p : The day after Friday is Sunday. q: 1 _ 2 r : If -4x - 2 = 10, then x = 3. = 0.5 1. r ⋀ q 4. q ⋀ ∼q 2. r ⋁ p 5. ∼q ⋁ q 3. p ⋁ r 6. q ⋁ r Construct a truth table for each compound statement. 7. s ⋀ ∼t 8. ∼u ⋁ t 9. ∼u ⋁ (s ⋀ t) Use a truth table to show that the two statements are logically equivalent. 10. p → q; ∼q → ∼p 11. q → p; ∼p → ∼q 12. A biconditional statement can be written as (p → q) ⋀ (q → p). Construct a truth table for this compound statement. 13. DeMorgan’s Laws state that ∼ (p ⋀ q) = ∼p ⋁ ∼q and that ∼ (p ⋁ q) = ∼p ⋀ ∼q. a. Use truth tables to show that both statements are true. b. If you think of disjunction and conjunction as inverse operations, DeMorgan’s Laws are similar to which algebraic property? 14. The Law of Disjunctive Inference states that if p ⋁ q

is true and p is false, then q must be true. a. Construct a truth table for p ⋁ q. b. Use the truth table to explain why the Law of Disjunctive Inference is true. Chapter 2 Extension 129 129 For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary biconditional statement...... 96 definition................... 97 paragraph proof............ 120 conclusion.................. 81 flowchart proof............. 118 polygon..................... 98 conditional statement........ 81 hypothesis.................. 81 proof...................... 104 conjecture.................. 74 inductive reasoning.......... 74 quadrilateral................ 98 contrapositive............... 83 inverse...................... 83 theorem................... 110 converse.................... 83 logically equivalent triangle..................... 98 counterexample............. 75 deductive reasoning......... 88 statements................ 83 negation.................... 82 truth value.................. 82 two-column proof..........

111 Complete the sentences below with vocabulary words from the list above. 1. A statement you can prove and then use as a reason in later proofs is a(n)?. ̶̶̶ 2.? is the process of using logic to draw conclusions from given facts, definitions, ̶̶̶ and properties. 3. A(n)? is a case in which a conjecture is not true. ̶̶̶ 4. A statement you believe to be true based on inductive reasoning is called a(n)?. ̶̶̶ 2-1 Using Inductive Reasoning to Make Conjectures (pp. 74–79) E X A M P L E S EXERCISES TEKS G.2.B, G.3.D, G.5.B ■ Find the next item in the pattern below. Make a conjecture about each pattern. Write the next two items. The red square moves in the counterclockwise direction. The next figure is. 5. ■ Complete the conjecture “The sum of two?.” ̶̶̶ odd numbers is List some examples and look for a pattern + 11 = 18 Complete each conjecture. 8. The sum of an even number and an odd number is?. ̶̶̶ The sum of two odd numbers is even. 9. The square of a natural number is?. ̶̶̶ ■ Show that the conjecture “For all non-zero integers, -x < x” is false by finding a counterexample. Pick positive and negative values for x and substitute to see if the conjecture holds. Let n = 3. Since -3 < 3, the conjecture holds. Let n = -5. Since - (-5) is 5 and 5 ≮ -5, the conjecture is false. n = -5 is a counterexample. 130 130 Chapter 2 Geometric Reasoning Determine if each conjecture is true. If not, write or draw a counterexample. 10. All whole numbers are natural numbers. ̶̶ BC. 11. If C is the midpoint of ̶̶ AB, then ̶̶ AC ≅ 12. If 2x + 3 = 15, then x = 6. 13. There are 28 days in February. 14. Draw a triangle. Construct the bisectors of each angle of the triangle. Make a conjecture about where the three angle bisectors intersect. �� 2-2 Conditional Statements (

pp. 81–87) TEKS G.3.A, G.3.C E X A M P L E S EXERCISES ■ Write a conditional statement from the sentence “A rectangle has congruent diagonals.” If a figure is a rectangle, then it has congruent diagonals. ■ Write the inverse, converse, and contrapositive of the conditional statement “If m∠1 = 35°, then ∠1 is acute.” Find the truth value of each. Converse: If ∠1 is acute, then m∠1 = 35°. Not all acute angles measure 35°, so this is false. Inverse: If m∠1 ≠ 35°, then ∠1 is not acute. You can draw an acute angle that does not measure 35°, so this is false. Contrapositive: If ∠1 is not acute, then m∠1 ≠ 35°. An angle that measures 35° must be acute. So this statement is true. Write a conditional statement from each Venn diagram. 15. 16. Determine if each conditional is true. If false, give a counterexample. 17. If two angles are adjacent, then they have a common ray. 18. If you multiply two irrational numbers, the product is irrational. Write the converse, inverse, and contrapositive of each conditional statement. Find the truth value of each. 19. If ∠X is a right angle, then m∠X = 90°. 20. If x is a whole number, then x = 2. 2-3 Using Deductive Reasoning to Verify Conjectures (pp. 88–93) TEKS G.2.B, E X A M P L E S EXERCISES G.3.B, G.3.C, G.3.E ■ Determine if the conjecture is valid by the Law of Detachment or the Law of Syllogism. Given: If 5c = 8y, then 2w = -15. If 5c = 8y, then x = 17. Conjecture: If 2w = -15, then x = 17. Let p be 5c = 8y, q be 2w = -15, and r be x = 17. Using symbols, the given information is written as p → q and p → r. Neither the Law of Detachment nor the Law

of Syllogism can be applied. The conjecture is not valid. Use the true statements below to determine whether each conclusion is true or false. Sue is a member of the swim team. When the team practices, Sue swims. The team begins practice when the pool opens. The pool opens at 8 A.M. on weekdays and at 12 noon on Saturday. 21. The swim team practices on weekdays only. 22. Sue swims on Saturdays. 23. Swim team practice starts at the same time every day. ■ Draw a conclusion from the given information. Given: If two points are distinct, then there is one line through them. A and B are distinct points. Let p be the hypothesis: two points are distinct. Let q be the conclusion: there is one line through the points. The statement “A and B are distinct points” matches the hypothesis, so you can conclude that there is one line through A and B. Use the following information for Exercises 24–26. The expression 2.15 + 0.07x gives the cost of a long-distance phone call, where x is the number of minutes after the first minute. If possible, draw a conclusion from the given information. If not possible, explain why. 24. The cost of Sara’s long-distance call is $2.57. 25. Paulo makes a long-distance call that lasts ten minutes. 26. Asa’s long-distance phone bill for the month is $19.05. Study Guide: Review 131 131 �� 2-4 Biconditional Statements and Definitions (pp. 96–101) TEKS G.3.A, G.3.B E X A M P L E S EXERCISES ■ For the conditional “If a number is divisible by 10, then it ends in 0”, write the converse and a biconditional statement. Converse: If a number ends in 0, then it is divisible by 10. Biconditional: A number is divisible by 10 if and only if it ends in 0. ■ Determine if the biconditional “The sides of a triangle measure 3, 7, and 15 if and only if the perimeter is 25” is true. If false, give a counterexample. Conditional: If the sides of a triangle measure 3, 7, and 15, then the perimeter is 25. True.

Converse: If the perimeter of a triangle is 25, then its sides measure 3, 7, and 15. False; a triangle with side lengths of 6, 10, and 9 also has a perimeter of 25. Therefore the biconditional is false. Determine if a true biconditional can be written from each conditional statement. If not, give a counterexample. 27. If 3 - 2x_ 5 = 2, then x = 5_. 2 28. If x < 0, then the value of x 4 is positive. 29. If a segment has endpoints at (1, 5) and (-3, 1), then its midpoint is (-1, 3). 30. If the measure of one angle of a triangle is 90°, then the triangle is a right triangle. Complete each statement to form a true biconditional. 31. Two angles are? if and only if the sum of ̶̶̶ their measures is 90°. 32. x 3 >0 if and only if x is?. ̶̶̶ 33. Trey can travel 100 miles in less than 2 hours if and only if his average speed is?. ̶̶̶ 34. The area of a square is equal to s 2 if and only if the perimeter of the square is?. ̶̶̶ 2-5 Algebraic Proof (pp. 104–109) TEKS G.3.B, G.3.C, G.3.E E X A M P L E S EXERCISES ■ Solve the equation 5x - 3 = -18. Write a justification for each step. 5x - 3 = -18 + 3 + 3 ̶̶̶ ̶̶̶̶̶ 5x = -15 -15_ 5x_ 5 5 x = -3 = Given Add. Prop. of = Simplify. Div. Prop. of = Simplify. ■ Write a justification for each step. RS = ST Given 5x - 18 = 4x x - 18 = 0 x = 18 Subst. Prop. of = Subtr. Prop. of = Add. Prop. of = Identify the property that justifies each statement. ■ ∠X ≅ ∠2, so ∠2 ≅ ∠X. Symmetric Property of Congruence ■ If m∠2 = 180° and m∠3 = 180°, then m∠2 = m∠

3. Transitive Property of Equality 132 132 Chapter 2 Geometric Reasoning Solve each equation. Write a justification for each step. 35. m_ -5 36. -47 = 3x - 59 + 3 = -4.5 Identify the property that justifies each statement. 37. a + b = a + b 38. If ∠RST ≅ ∠ABC, then ∠ABC ≅ ∠RST. 39. 2x = 9, and y = 9. So 2x = y. Use the indicated property to complete each statement. 40. Reflex. Prop. of ≅: figure ABCD ≅? ̶̶̶ 41. Sym. Prop. of =: If m∠2 = m∠5, then ̶̶ CD and ̶̶ AB ≅?. ̶̶̶ ̶̶ EF, 42. Trans. Prop. of ≅: If?. ̶̶̶ then ̶̶ AB ≅ 43. Kim borrowed money at an annual simple interest rate of 6% to buy a car. How much did she borrow if she paid $4200 in interest over the life of the 4-year loan? Solve the equation I = Prt for P and justify each step. �� 2-6 Geometric Proof (pp. 110–116) TEKS G.1.A, G.3.B, G.3.C, G.3.E E X A M P L E S EXERCISES ■ Write a justification for each step, given that m∠2 = 2m∠1. 1. ∠1 and ∠2 supp. 2. m∠1 + m∠2 = 180° 3. m∠2 = 2m∠1 4. m∠1 + 2m∠1 = 180° 5. 3m∠1 = 180° 6. m∠1 = 60° Lin. Pair Thm. Def. of supp.  Given Subst. Steps 2, 3 Simplify Div. Prop. of = ■ Use the given plan to write a two-column proof. Given: ̶̶ AD bisects ∠BAC. ∠1 ≅ ∠3 Prove: ∠2 ≅ ∠3 Plan: Use the definition of angle bisector to show that ∠1 ≅ ∠2. Use the Transitive Property to conclude that �

�2 ≅ ∠3. Two-column proof: Statements Reasons ̶̶̶ AD bisects ∠BAC. 1. 1. Given 2. ∠1 ≅ ∠2 3. ∠1 ≅ ∠3 4. ∠2 ≅ ∠3 2. Def. of ∠ bisector 3. Given 4. Trans. Prop. of ≅ 44. Write a justification for each step, given that ∠1 and ∠2 are complementary, and ∠1 ≅ ∠3. 1. ∠1 and ∠2 comp. 2. m∠1 + m∠2 = 90° 3. ∠1 ≅ ∠3 4. m∠1 = m∠3 5. m∠3 + m∠2 = 90° 6. ∠3 and ∠2 comp. 45. Fill in the blanks to complete the two-column proof. ̶̶ TU ≅ Given: Prove: SU + TU = SV Two-column proof: ̶̶ UV Statements Reasons ̶̶ TU ≅ ̶̶ UV 1. 2. b. 3. c.? ̶̶̶̶? ̶̶̶̶ 4. SU + TU = SV 1. a.? ̶̶̶̶ 2. Def. of ≅ segs. 3. Seg. Add. Post. 4. d.? ̶̶̶̶ Find the value of each variable. 46. 47. 2-7 Flowchart and Paragraph Proofs (pp. 118–125) TEKS G.1.A, G.2.B, G.3.C, G.3.E E X A M P L E S EXERCISES Use the two-column proof in the example for Lesson 2-6 above to write each of the following. ■ a flowchart proof ■ a paragraph proof ̶̶ AD bisects ∠BAC, ∠1 ≅ ∠2 by the Since definition of angle bisector. It is given that ∠1 ≅ ∠3. Therefore, ∠2 ≅ ∠3 by the Transitive Property of Congruence. Use the given plan to write each of the following. Given: ∠ADE and ∠DAE are complementary. ∠ADE and ∠BAC are complementary. Prove: ∠DAC �

� ∠BAE Plan: Use the Congruent Complements Theorem to show that ∠DAE ≅ ∠BAC. Since ∠CAE ≅ ∠CAE, ∠DAC ≅ ∠BAE by the Common Angles Theorem. 48. a flowchart proof 49. a paragraph proof Find the value of each variable and name the theorem that justifies your answer. 50. 51. Study Guide: Review 133 133 �� Find the next item in each pattern. 1. 2. 405, 135, 45, 15, … 3. Complete the conjecture “The sum of two even numbers is?. ” ̶̶̶ 4. Show that the conjecture “All complementary angles are adjacent” is false by finding a counterexample. 5. Identify the hypothesis and conclusion of the conditional statement “The show is cancelled if it rains.” 6. Write a conditional statement from the sentence “Parallel lines do not intersect.” Determine if each conditional is true. If false, give a counterexample. 7. If two lines intersect, then they form four right angles. 8. If a number is divisible by 10, then it is divisible by 5. Use the conditional “If you live in the United States, then you live in Kentucky” for Items 9–11. Write the indicated type of statement and determine its truth value. 9. converse 10. inverse 11. contrapositive 12. Determine if the following conjecture is valid by the Law of Detachment. Given: If it is colder than 50°F, Tom wears a sweater. It is 46°F today. Conjecture: Tom is wearing a sweater. 13. Use the Law of Syllogism to draw a conclusion from the given information. Given: If a figure is a square, then it is a quadrilateral. If a figure is a quadrilateral, then it is a polygon. Figure ABCD is a square. 14. Write the conditional statement and converse within the biconditional “Chad will work on Saturday if and only if he gets paid overtime.” 15. Determine if the biconditional “B is the midpoint of ̶̶ AC iff

AB = BC” is true. If false, give a counterexample. Solve each equation. Write a justification for each step. 16. 8 - 5s = 1 17. 0.4t + 3 = 1.6 18. 38 = -3w + 2 Identify the property that justifies each statement. 19. If 2x = y and y = 7, then 2x = 7. 21. ∠X ≅ ∠P, and ∠P ≅ ∠D. So ∠X ≅ ∠D. Use the given plan to write a proof in each format. 20. m∠DEF = m∠DEF ̶̶ XY ≅ ̶̶ XY, then ̶̶ ST ≅ 22. If ̶̶ ST.  FB bisects ∠AFC. Given: ∠AFB ≅ ∠EFD Prove: Plan: Since vertical angles are congruent, ∠EFD ≅ ∠BFC. Use the Transitive Property to conclude that ∠AFB ≅ ∠BFC. Thus  FB bisects ∠AFC by the definition of angle bisector. 23. two-column proof 24. paragraph proof 25. flowchart proof 134 134 Chapter 2 Geometric Reasoning �� FOCUS ON SAT MATHEMATICS SUBJECT TESTS Some colleges require that you take the SAT Subject Tests. There are two math subject tests—Level 1 and Level 2. Take the Mathematics Subject Test Level 1 when you have completed three years of college-prep mathematics courses. You may want to time yourself as you take this practice test. It should take you about 6 minutes to complete. On SAT Mathematics Subject Test questions, you receive one point for each correct answer, but you lose a fraction of a point for each incorrect response. Guess only when you can eliminate at least one of the answer choices. 1. In the figure below, m∠1 = m∠2. What is the value of y? 3. What is the contrapositive of the statement “If it is raining, then the football team will win”? Note: Figure not drawn to scale. (A) 10 (C) 40 (E) 60 (B) 30 (D) 50 2. The statement “I will cancel my appointment if and only if I have

a conflict” is true. Which of the following can be concluded? I. If I have a conflict, then I will cancel my appointment. II. If I do not cancel my appointment, then I do not have a conflict. III. If I cancel my appointment, then I have a conflict. (A) I only (C) III only (E) I, II, and III (B) II only (D) I and III (A) If it is not raining, then the football team will not win. (B) If it is raining, then the football team will not win. (C) If the football team wins, then it is raining. (D) If the football team does not win, then it is not raining. (E) If it is not raining, then the football team will win. 4. Given the points D (1, 5) and E (-2, 3), which conclusion is NOT valid? ̶̶ (A) The midpoint of DE is (- 1 _, 4). 2 (B) D and E are collinear. (C) The distance between D and E is √  5. (D) ̶̶ DE ≅ ̶̶ ED (E) D and E are distinct points. 5. For all integers x, what conclusion can be drawn about the value of the expression x 2 __ 2? (A) The value is negative. (B) The value is not negative. (C) The value is even. (D) The value is odd. (E) The value is not a whole number. College Entrance Exam Practice 135 135 �� Gridded Response: Record Your Answer When responding to a gridded-response test item, you must fill out the grid on your answer sheet correctly, or the item will be marked as incorrect. Gridded Response: Solve the equation 1258 - 2 (3x - 72) = -80. The value of x is 247. • Using a pencil, write your answer in the answer boxes at the top of the grid. • Put only one digit in each box. The decimal point has a designated column. • Do not leave a blank box in the middle of an answer. • For each digit, shade the bubble that is in the same column as the digit in the answer box. Gridded Response: The perimeter of a rectangle is 90 in. The width of the rectangle is 18 in

. Find the length of the rectangle in feet. The length of the rectangle is 27 inches, but the problem asks for the measurement in feet. 27 inches = 9 _ 4 • Fractions and mixed numbers cannot be gridded, so you must grid, or 2.25, feet the answer as 2.25. • Using a pencil, write your answer in the answer boxes at the top of the grid. • Put only one digit in each box. The decimal point has a designated column. • Do not leave a blank box in the middle of an answer. • For each digit, shade the bubble that is in the same column as the digit in the answer box. 136 136 Chapter 2 Geometric Reasoning �� Sample C The length of a segment is 897 2 __ gridded this answer as shown. units. A student 5 �� You cannot grid a negative number in a griddedresponse item because the grid does not include the negative sign (-). So if you get a negative answer to a test item, rework the problem. You probably made a math error. Read each statement and answer the questions that follow. Sample A The correct answer to a test item is 1.6. A student gridded this answer as shown. 5. What answer does the grid show? 6. Explain why you cannot grid a fraction or a mixed number. 7. Write the answer 897 2__ 5 be entered in the grid correctly. in a form that could 1. What error did the student make when filling out the grid? 2. Another student got an answer of -1.6. Explain why the student knew this answer was wrong. 8. Another student got an answer of 10,216.5 units. Explain why the student knew this answer was wrong. Sample D The measure of an angle is 48.9°. A student gridded this answer as shown. Sample B The perimeter of a triangle is 2 3 __ gridded this answer as shown. feet. A student 4 9. What answer does the grid show? 10. What error did the student make when filling out the grid? 11. Explain how to correctly grid the answer. 3. What error did the student make when filling out the grid? 4. Explain how to correctly grid the answer. TAKS Tackler 137 137

�� KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1–2 Multiple Choice Use the figure below for Items 1 and 2. In the figure,   DB bisects ∠ADC. 5. A diagonal of a polygon connects nonconsecutive vertices. The table shows the number of diagonals in a polygon with n sides. Number of Sides Number of Diagonals 4 5 6 7 2 5 9 14 If the pattern continues, how many diagonals does a polygon with 8 sides have? 17 19 20 21 6. Which type of transformation maps figure LMNP onto figure L’M’N’P’? 1. Which best describes the intersection of ∠ADB and ∠BDC? Exactly one ray Exactly one point Exactly one angle Exactly one segment 2. Which expression is equal to the measure of ∠ADC? 2 (m∠ADB) 90° - m∠BDC 180° - 2 (m∠ADC) m∠BDC - m∠ADB 3. What is the inverse of the statement, “If a polygon has 8 sides, then it is an octagon”? Reflection Rotation Translation None of these If a polygon is an octagon, then it has 8 sides. If a polygon is not an octagon, then it does not have 8 sides. If an octagon has 8 sides, then it is a polygon. If a polygon does not have 8 sides, then it is not an octagon. 4. Lily conjectures that if a number is divisible by 15, then it is also divisible by 9. Which of the following is a counterexample? 45 50 60 72 7. Miyoko went jogging on July 25, July 28, July 31, and August 3. If this pattern continues, when will Miyoko go jogging next? August 5 August 6 August 7 August 8 8. Congruent segments have equal measures. A segment bisector

divides a segment into ̶̶ two congruent segments. DE at X  XY intersects ̶̶ DE. Which conjecture is valid? and bisects m∠YXD = m∠YXE Y is between D and E. DX = XE DE = YE 138 138 Chapter 2 Geometric Reasoning �� 9. Which statement is true by the Symmetric Property of Congruence? ̶̶ ST ̶̶ ST ≅ 15 + MN = MN + 15 If ∠P ≅ ∠Q, then ∠Q ≅ ∠P. If ∠D ≅ ∠E and ∠E ≅ ∠F, then ∠D ≅ ∠F. �� To find a counterexample for a biconditional statement, write the conditional statement and converse it contains. Then try to find a counterexample for one of these statements. 10. Which is a counterexample for the following biconditional statement? A pair of angles is supplementary if and only if the angles form a linear pair. The measures of supplementary angles add to 180°. A linear pair of angles is supplementary. Complementary angles do not form a linear pair. Two supplementary angles are not adjacent. 11. K is between J and L. The distance between J and K is 3.5 times the distance between K and L. If JK = 14, what is JL? 10.5 18 24.5 49 STANDARDIZED TEST PREP Short Response 16. Solve the equation 2 (AB) + 16 = 24 to find the length of segment AB. Write a justification for each step. 17. Use the given two-column proof to write a flowchart proof. ̶̶ Given: DE ≅ Prove: DE = FG + GH ̶̶ FH Two-column proof: Statements ̶̶ FH ̶̶ DE ≅ 1. Reasons 1. Given 2. DE = FH 2. Def. of ≅ segs. 3. FG + GH = FH 3. Seg. Add. Post. 4. DE = FG + GH 4. Subst. 18. Consider the following conditional statement. If two angles are complementary, then the angles are acute. a. Determine if the conditional is true or false. If false, give a counterexample. b. Write the

converse of the conditional statement. 12. What is the length of the segment connecting the points (-7, -5) and (5, -2)? c. Determine whether the converse is true or false. If false, give a counterexample. √  13 √  53 3 √  17 √  193 Gridded Response 13. A segment has an endpoint at (5, -2). The midpoint of the segment is (2, 2). What is the length of the segment? 14. ∠P measures 30° more than the measure of its supplement. What is the measure of ∠P in degrees? 15. The perimeter of a square field is 1.6 kilometers. What is the area of the field in square kilometers? Extended Response 19. The figure below shows the intersection of two lines. a. Name the linear pairs of angles in the figure. What conclusion can you make about each pair? Explain your reasoning. b. Name the pairs of vertical angles in the figure. What conclusion can you make about each pair? Explain your reasoning. c. Suppose m∠EBD = 90°. What are the measures of the other angles in the figure? Write a two-column proof to support your answer. Cumulative Assessment, Chapters 1–2 139 139 �� T E X A S TAKS Grades 9–11 Obj. 10 �� The Freescale Marathon Every February, runners take to the streets of Austin to participate in a 26-mile marathon. The course travels mostly downhill, with a net drop in elevation of more than 400 feet from beginning to end. But don’t be fooled; many former participants strongly recommend hill training for this course! Choose one or more strategies to solve each problem. 1. During the marathon, a runner maintains a steady pace and completes the first 2.6 miles in 20 minutes. After 1 hour 20 minutes, she has completed 10.4 miles. Make a conjecture about the runner’s average speed in miles per hour. How long do you expect her to take to complete the marathon? 2. The course features fully equipped aid stations with medical support for the complete eight-hour duration of the race. From mile 2 to mile 6, these stations are located every other mile. After that, they are located every mile to the finish. Portable toilets are available at

each mile marker and at the end of the course. At how many points are there both an aid station and portable toilets? �� For 3, use the map. 3. The course includes a straight section along Forty-fifth Street from Shoal Creek Boulevard to Duval Street. The distance from Guadalupe Street to Duval is twice the distance from Burnet Road to Lamar Boulevard. The distance from Lamar to Guadalupe is 210 feet greater than the distance from Shoal Creek to Burnet. What is the distance from Guadalupe to Duval? 140 140 Chapter 2 Geometric Reasoning � � � � � � � � � � � � � � � � � � � � �� Show Caves The region between San Antonio and Austin, known as the Texas Hill Country, is home to six of the seven show caves in the state. A show cave is a cave developed for public use, typically with amenities such as lighting and groomed trails. The seventh show cave in Texas, the Caverns of Sonora, is internationally recognized as one of the most beautiful in the world. Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List Choose one or more strategies to solve each problem. 1. Jared took a tour at least 5 __ 8 mi long and saw a cave that is less than 10,000 ft in length. Which caves might Jared have visited? 2. A travel brochure includes the following statements about Texas show caves. Determine whether each statement is true or false. If false, explain why. a. If you tour a cave that is more Texas Show Caves Cave Length (ft) Approximate Depth (ft) Tour Length (mi) Cascade Caverns Cave Without a Name Caverns of Sonora Inner Space Cavern Longhorn Cavern Natural Bridge Caverns Wonder Cave 1,700 14,211 20,000 15,000 9,850 8,600 1,296 132 89 150 80 23 250 91 0.25 0.

25 1.5 1.2 0.625 0.75 0.08 than 100 ft deep, then you’ll see a cave that is more than 8000 ft in length. b. If you haven’t been to the Caverns of Sonora, then you haven’t seen a cave that is at least 15,000 ft long. c. If you don’t want to walk more than a mile, but you want to see a cave with a depth of at least 150 ft, then you should visit Natural Bridge Caverns. 3. Inner Space Cavern has a second tour, which Ingrid completes in 18 min and 45 s. If Ingrid walks 12,672 ft in 1 h, what is the length of the second tour? Problem Solving on Location 141141 Parallel and Perpendicular Lines 3A Lines with Transversals 3-1 Lines and Angles Lab Explore Parallel Lines and Transversals 3-2 Angles Formed by Parallel Lines and Transversals 3-3 Proving Lines Parallel Lab Construct Parallel Lines 3-4 Perpendicular Lines Lab Construct Perpendicular Lines 3B Coordinate Geometry 3-5 Slopes of Lines Lab Explore Parallel and Perpendicular Lines 3-6 Lines in the Coordinate Plane KEYWORD: MG7 ChProj Sailboats at the Corpus Christi Municipal Marina 142 142 Chapter 3 Vocabulary Match each term on the left with a definition on the right. 1. acute angle A. segments that have the same length 2. congruent angles B. an angle that measures greater than 90° and less than 180° 3. obtuse angle 4. collinear C. points that lie in the same plane D. angles that have the same measure 5. congruent segments E. points that lie on the same line F. an angle that measures greater than 0° and less than 90° Conditional Statements Identify the hypothesis and conclusion of each conditional. 6. If E is on AC, then E lies in plane P. 7. If A is not in plane Q, then A is not on BD. 8. If plane P and plane Q intersect, then they intersect in a line. Name and Classify Angles Name and classify each angle. 9. 10. 11. 12. Angle Relationships Give an example of each angle pair. 13. vertical angles 14. adjacent angles 15.

complementary angles 16. supplementary angles Evaluate Expressions Evaluate each expression for the given value of the variable. 17. 4x + 9 for x = 31 18. 6x - 16 for x = 43 19. 97 - 3x for x = 20 20. 5x + 3x + 12 for x = 17 Solve Multi-Step Equations Solve each equation for x. 21. 4x + 8 = 24 23. 4x + 3x + 6 = 90 22. 2 = 2x - 8 24. 21x + 13 + 14x - 8 = 180 Parallel and Perpendicular Lines 143 143 �� Key Vocabulary/Vocabulario alternate exterior angles alternate interior angles corresponding angles parallel lines ángulos alternos externos ángulos alternos internos ángulos correspondientes líneas paralelas perpendicular bisector mediatriz perpendicular lines líneas perpendiculares same-side interior angles ángulos internos del mismo lado slope transversal pendiente transversal Geometry TEKS G.1.A Geometric Structure* develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, answer the following questions. You may refer to the chapter, the glossary, or a dictionary if you like. 1. The root trans- means “across.” What do you think a transversal of two lines does? 2. The slope of a mountain trail describes the steepness of the climb. What might the slope of a line describe? 3. What does the word corresponding mean? What do you think the term corresponding angles means? 4. What does the word interior mean? What might the phrase “interior of a pair of lines” describe? The word alternate means “to change from one to another.” If two lines are crossed by a third line, where do you think a pair of alternate interior angles might be? 3-2 Tech. Lab Les. 3-1 Les. 3-2 Les. 3-3 ★ 3-3 Geo. Lab Les. 3-4 ★ 3-4 Geo. Lab 3-6 Tech. Lab Les. 3-5 Les. 3-6 G.2.A Geometric Structure* use construction to ★

★ ★ explore attributes of geometric figures and to make conjectures about geometric relationships G.3.C Geometric Structure* use logical reasoning to ★ ★ ★ ★ prove statements are true... G.7.B Dimensionality and the geometry of location* use slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines... G.7.C Dimensionality and the geometry of location* develop and use formulas involving... slope... ★ ★ ★ ★ G.9.A Congruence and the geometry of ★ ★ ★ ★ ★ ★ size* formulate and test conjectures about the properties of parallel and perpendicular lines based on explorations * Knowledge and skills are written out completely on pages TX28–TX35. 144 144 Chapter 3 Study Strategy: Take Effective Notes Taking effective notes is an important study strategy. The Cornell system of note taking is a good way to organize and review main ideas. In the Cornell system, the paper is divided into three main sections. The note-taking column is where you take notes during lecture. The cue column is where you write questions and key phrases as you review your notes. The summary area is where you write a brief summary of the lecture. Step 2: Cues After class, write down key phrases or questions in the left column. Step 3: Summary Use your cues to restate the main points in your own words. 9/4/05 Chapter 2 Lesson 6 page 1 What can you use to just ify steps in a proof? Geometric proof: Start with hypothes is, and then use defs., and thms. to reach posts. conc lus ion. Just ify each step., Step 1: Notes Draw a vertical line about 2.5 inches from the left side of your paper. During class, write your notes about the main points of the lecture in the right column. What k ind of ang les form a l inear pa ir? Linear Pa ir Theorem If 2 ∠ s form a l in. pa ir, then they are supp. Congruent Supp lements Theorem If 2 ∠ s are supp. to the same ∠ (or to 2 ≅ ∠ s), then the 2 ∠ s are ≅. What is true about two supp lements of the same ang le? S u m m a ry : def i n i t i on s, p o stu l ate s, a n d th e o re m s to sh ow th at a c

on c l u s i on i s tr u e. Th e Li n e a r Pa i r Th e o re m s ay s th at two a n g le s th at fo re s u p p le m ent a ry. Th e C on g r u ent S u p p le m ent s Th e o re m s ay s th at two s u p p le m ent s to th e s a m e a n g le a re c on g r u ent. Try This 1. Research and write a paragraph describing the Cornell system of note taking. Describe how you can benefit from using this type of system. 2. In your next class, use the Cornell system of note taking. Compare these notes to your notes from a previous lecture. Parallel and Perpendicular Lines 145 145 3-1 Lines and Angles Objectives Identify parallel, perpendicular, and skew lines. Identify the angles formed by two lines and a transversal. Vocabulary parallel lines perpendicular lines skew lines parallel planes transversal corresponding angles alternate interior angles alternate exterior angles same-side interior angles Who uses this? Card architects use playing cards to build structures that contain parallel and perpendicular planes. Bryan Berg uses cards to build structures like the one at right. In 1992, he broke the Guinness World Record for card structures by building a tower 14 feet 6 inches tall. Since then, he has built structures more than 25 feet tall. Parallel, Perpendicular, and Skew Lines Parallel lines (ǁ) are coplanar and do not intersect. In the figure,   AB ǁ   EF, and   EG ǁ   FH. Perpendicular lines (⊥) intersect at 90° angles. In the figure,   AB ⊥   AE, and   EG ⊥   GH. Skew lines are not coplanar. Skew lines are not parallel and do not intersect. In the figure,   AB and   EG are skew. Parallel planes are planes that do not intersect. In the figure, plane ABE �

� plane CDG. E X A M P L E 1 Identifying Types of Lines and Planes Identify each of the following. Arrows are used to show that   AB ǁ   EF and   EG ǁ   FH. Segments or rays are parallel, perpendicular, or skew if the lines that contain them are parallel, perpendicular, or skew. A a pair of parallel segments ̶̶ KN ǁ ̶̶ PS B a pair of skew segments ̶̶ RS are skew. ̶̶̶ LM and C a pair of perpendicular segments ̶̶̶ MR ⊥ ̶̶ RS D a pair of parallel planes plane KPS ǁ plane LQR Identify each of the following. 1a. a pair of parallel segments 1b. a pair of skew segments 1c. a pair of perpendicular segments 1d. a pair of parallel planes 146 146 Chapter 3 Parallel and Perpendicular Lines �� Angle Pairs Formed by a Transversal TERM EXAMPLE A transversal is a line that intersects two coplanar lines at two different points. The transversal t and the other two lines r and s form eight angles. Corresponding angles lie on the same side of the transversal t, on the same sides of lines r and s. Alternate interior angles are nonadjacent angles that lie on opposite sides of the transversal t, between lines r and s. Alternate exterior angles lie on opposite sides of the transversal t, outside lines r and s. Same-side interior angles or consecutive interior angles lie on the same side of the transversal t, between lines r and s. ∠1 and ∠5 ∠3 and ∠6 ∠1 and ∠8 ∠3 and ∠5 E X A M P L E 2 Classifying Pairs of Angles Give an example of each angle pair. A corresponding angles B alternate interior angles ∠4 and ∠8 ∠4 and ∠6 C alternate exterior angles D same-side interior angles ∠2 and ∠8 ∠4 and ∠5 Give an example of each angle pair. 2a. corresponding angles 2b. alternate interior angles 2c. alternate exterior angles 2d. same-side interior angles E

X A M P L E 3 Identifying Angle Pairs and Transversals Identify the transversal and classify each angle pair. To determine which line is the transversal for a given angle pair, locate the line that connects the vertices. A ∠1 and ∠5 transversal: n; alternate interior angles B ∠3 and ∠6 transversal: m; corresponding angles C ∠1 and ∠4 transversal: ℓ; alternate exterior angles 3. Identify the transversal and classify the angle pair ∠2 and ∠5 in the diagram above. 3- 1 Lines and Angles 147 147 �� THINK AND DISCUSS 1. Compare perpendicular and intersecting lines. 2. Describe the positions of two alternate exterior angles formed by lines m and n with transversal p. 3. GET ORGANIZED Copy the diagram and graphic organizer. In each box, list all the angle pairs of each type in the diagram. 3-1 Exercises Exercises KEYWORD: MG7 3-1 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary? are located on opposite sides of a transversal, between the two ̶̶̶̶ lines that intersect the transversal. (corresponding angles, alternate interior angles, alternate exterior angles, or same-side interior angles Identify each of the following. p. 146 2. one pair of perpendicular segments 3. one pair of skew segments 4. one pair of parallel segments 5. one pair of parallel planes Give an example of each angle pair. p. 147 6. alternate interior angles 7. alternate exterior angles 8. corresponding angles 9. same-side interior angles Identify the transversal and classify each angle pair. p. 147 10. ∠1 and ∠2 11. ∠2 and ∠3 12. ∠2 and ∠4 13. ∠4 and ∠5 148 148 Chapter 3 Parallel and Perpendicular Lines �� Independent Practice For See Exercises Example 14–17 18–21 22–25 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S8 Application Practice p. S30 PRACTICE AND PROBLEM

SOLVING Identify each of the following. 14. one pair of parallel segments 15. one pair of skew segments 16. one pair of perpendicular segments 17. 17. one pair of parallel planes Give an example of each angle pair. 18. same-side interior angles 19. alternate exterior angles 20. corresponding angles 21. alternate interior angles Identify the transversal and classify each angle pair. 22. ∠2 and ∠3 23. ∠4 and ∠5 24. ∠2 and ∠4 25. ∠1 and ∠2 26. Sports A football player runs across the 30-yard line at an angle. He continues in a straight line and crosses the goal line at the same angle. Describe two parallel lines and a transversal in the diagram. Name the type of angle pair shown in each letter. 27. F 28. Z 29. C Entertainment Entertainment In an Ames room, two people of the same height that are standing in different parts of the room appear to be different sizes. Entertainment Entertainment Use the following information for Exercises 30–32. In an Ames room, the floor is tilted and the back wall is closer to the front wall on one side. 30. 30. Name a pair of parallel segments in the diagram. 31. Name a pair of skew segments in the diagram. 32. Name a pair of perpendicular segments in the diagram. 3- 1 Lines and Angles 149 149 �� 33. This problem will prepare you for the Multi-Step TAKS Prep on p 180. Buildings that are tilted like the one shown are sometimes called mystery spots. a. Name a plane parallel to plane KLP, a plane parallel to plane KNP, and a plane parallel to KLM. b. In the diagram, ̶̶ QR is a transversal to ̶̶ PQ and ̶̶ RS. What type of angle pair is ∠PQR and ∠QRS? 34. Critical Thinking Line ℓ is contained in plane P and line m is contained in plane Q. If P and Q are parallel, what are the possible classifications of ℓ and m? Include diagrams to support your answer. Use the diagram for Exercises 35–40. 35. Name a pair of alternate interior angles with transversal n. 36. Name a pair of same-side interior angles with transversal �

��. 37. Name a pair of corresponding angles with transversal m. 38. Identify the transversal and classify the angle pair for ∠3 and ∠7. 39. Identify the transversal and classify the angle pair for ∠5 and ∠8. 40. Identify the transversal and classify the angle pair for ∠1 and ∠6. 41. Aviation Describe the type of lines formed by two planes when flight 1449 is flying from San Francisco to Atlanta at 32,000 feet and flight 2390 is flying from Dallas to Chicago at 28,000 feet. 42. Multi-Step Draw line p, then draw two lines m and n that are both perpendicular to p. Make a conjecture about the relationship between lines m and n. 43. Write About It Discuss a real-world example of skew lines. Include a sketch. 44. Which pair of angles in the diagram are alternate interior angles? ∠1 and ∠5 ∠2 and ∠6 ∠7 and ∠5 ∠2 and ∠3 45. How many pairs of corresponding angles are in the diagram? 2 4 8 16 150 150 Chapter 3 Parallel and Perpendicular Lines ��San FranciscoAtlantaChicagoDallasHolt, Rinehart & WinstonHigh School Mathge07sec03101007a Locator map showing cities3rd proof�� 46. Which type of lines are NOT represented in the diagram? Parallel lines Skew lines Intersecting lines Perpendicular lines 47. For two lines and a transversal, ∠1 and ∠8 are alternate exterior angles, and ∠1 and ∠5 are corresponding angles. Classify the angle pair ∠5 and ∠8. Vertical angles Alternate interior angles Adjacent angles Same-side interior angles 48. Which angles in the diagram are NOT corresponding angles? ∠1 and ∠5 ∠2 and ∠6 ∠4 and ∠8 ∠2 and ∠7 � � � � � � � � CHALLENGE AND EXTEND Name all the angle pairs of each type in the diagram. Identify the transversal for each pair. 49. corresponding 50. alternate interior 51. alternate exterior 52. same-side interior 53. Multi-Step Draw two lines and a transversal such that ∠1 and ∠3 are corresponding angles, ∠1 and ∠2 are alternate interior angles,

and ∠3 and ∠4 are alternate exterior angles. What type of angle pair is ∠2 and ∠4? 54. If the figure shown is folded to form a cube, which faces of the cube will be parallel? � � � � � � � � � � � �� SPIRAL REVIEW Evaluate each function for x = -1, 0, 1, 2, and 3. (Previous course) 55. y = 4 x 2 - 7 56. y = -2 x 2 + 5 57. y = (x + 3) (x - 3) Find the circumference and area of each circle. Use the π key on your calculator and round to the nearest tenth. (Lesson 1-5) 58. �� 59. �� Write a justification for each statement, given that ∠1 and ∠3 are right angles. (Lesson 2-6) 60. ∠1 ≅ ∠3 61. m∠1 + m∠2 = 180° 62. ∠2 ≅ ∠4 � � � � 3- 1 Lines and Angles 151 151 Systems of Equations Algebra Sometimes angle measures are given as algebraic expressions. When you know the relationship between two angles, you can write and solve a system of equations to find angle measures. See Skills Bank page S67 Solving Systems of Equations by Using Elimination Step 1 Write the system so that like terms are under one another. Step 2 Eliminate one of the variables. Step 3 Substitute that value into one of the original equations and solve. Step 4 Write the answers as an ordered pair, (x, y). Step 5 Check your solution. Example 1 Solve for x and y. Since the lines are perpendicular, all of the angles are right angles. To write two equations, you can set each expression equal to 90°. (3x + 2y)° = 90°, (6x - 2y)° = 90° Step 1 Step 2 3x + 2y = 90 6x - 2y = 90 ̶̶̶̶̶̶̶̶ 9x + 0 = 180 Write the system so that like terms are under one another. Add like terms on each side of the equations. The y-term has been eliminated. x = 20 Divide both sides by 9 to solve for x. Step 3 3x + 2y = 90 Write

one of the original equations. 3 (20) + 2y = 90 Substitute 20 for x. 60 + 2y = 90 Simplify. 2y = 30 y = 15 Subtract 60 from both sides. Divide by 2 on both sides. Step 4 (20, 15) Write the solution as an ordered pair. Step 5 Check the solution by substituting 20 for x and 15 for y in the original equations. 3x + 2y = 90 6x - 2y = 90 3(20) + 2(15) 90 6(20) - 2(15) 90 60 + 30 90 120 - 30 90 90 90 ✓ 90 90 ✓ In some cases, before you can do Step 1 you will need to multiply one or both of the equations by a number so that you can eliminate a variable. 152 152 Chapter 3 Parallel and Perpendicular Lines �� Example 2 Solve for x and y. (2x + 4y)° = 72° (5x + 2y)° = 108° Vertical Angles Theorem Linear Pair Theorem The equations cannot be added or subtracted to eliminate a variable. Multiply the second equation by -2 to get opposite y-coefficients. 5x + 2y = 108 → -2 (5x + 2y) = -2 (108) → -10x - 4y = -216 Step 1 Step 2 2x + 4y = 72 -10x - 4y = -216 ̶̶̶̶̶̶̶̶̶̶̶̶ = -144 -8x Write the system so that like terms are under one another. Add like terms on both sides of the equations. The y-term has been eliminated. x = 18 Divide both sides by -8 to solve for x. Step 3 2x + 4y = 72 Write one of the original equations. 2(18) + 4y = 72 Substitute 18 for x. 36 + 4y = 72 Simplify. 4y = 36 y = 9 Subtract 36 from both sides. Divide by 4 on both sides. Step 4 (18, 9) Write the solution as an ordered pair. Step 5 Check the solution by substituting 18 for x and 9 for y in the original equations. 2x + 4y = 72 3(18) + 4(9) 36 + 36 72 72 5x + 2y = 108 5 (18) + 2 (9) 108 90 + 18 108 72 72 ✓ 108 108

✓ Try This TAKS Grades 9–11 Obj. 2, 4, 6 Solve for x and y. 1. 3. 2. 4. On Track for TAKS 153 153 �� 3-2 Use with Lesson 3-2 Activity Explore Parallel Lines and Transversals Geometry software can help you explore angles that are formed when a transversal intersects a pair of parallel lines. TEKS G.9.A Congruence and the geometry of size: formulate and test conjectures about the properties of parallel and perpendicular lines based on explorations.... KEYWORD: MG7 Lab3 1 Construct a line and label two points on the line A and B. 2 Create point C not on AB. Construct a line parallel to   AB through point C. Create another point on this line and label it D. 3 Create two points outside the two parallel lines and label them E and F. Construct transversal   EF. Label the points of intersection G and H. 4 Measure the angles formed by the parallel lines and the transversal. Write the angle measures in a chart like the one below. Drag point E or F and chart with the new angle measures. What relationships do you notice about the angle measures? What conjectures can you make? ∠AGE ∠BGE ∠AGH ∠BGH ∠CHG ∠DHG ∠CHF ∠DHF Angle Measure Measure Try This 1. Identify the pairs of corresponding angles in the diagram. What conjecture can you make about their angle measures? Drag a point in the figure to confirm your conjecture. 2. Repeat steps in the previous problem for alternate interior angles, alternate exterior angles, and same-side interior angles. 3. Try dragging point C to change the distance between the parallel lines. What happens to the angle measures in the figure? Why do you think this happens? 154 154 Chapter 3 Parallel and Perpendicular Lines 3-2 Angles Formed by Parallel Lines and Transversals TEKS G.3.C Geometric structure: use logical reasoning to prove statements are true.... Also G.3.E, G.9.A Objective Prove and use theorems about the angles formed by parallel lines and

a transversal. Who uses this? Piano makers use parallel strings for the higher notes. The longer strings used to produce the lower notes can be viewed as transversals. (See Example 3.) When parallel lines are cut by a transversal, the angle pairs formed are either congruent or supplementary. Postulate 3-2-1 Corresponding Angles Postulate THEOREM HYPOTHESIS CONCLUSION If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. ∠1 ≅ ∠3 ∠2 ≅ ∠4 ∠5 ≅ ∠7 ∠6 ≅ ∠8 E X A M P L E 1 Using the Corresponding Angles Postulate Find each angle measure. A m∠ABC x = 80 m∠ABC = 80° B m∠DEF Corr.  Post. (2x - 45) ° = (x + 30) ° Corr.  Post. x - 45 = 30 x = 75 m∠DEF = x + 30 Subtract x from both sides. Add 45 to both sides. = 75 + 30 = 105° Substitute 75 for x. 1. Find m∠QRS. Remember that postulates are statements that are accepted without proof. Since the Corresponding Angles Postulate is given as a postulate, it can be used to prove the next three theorems. 3- 2 Angles Formed by Parallel Lines and Transversals 155 155 �� Theorems Parallel Lines and Angle Pairs THEOREM HYPOTHESIS CONCLUSION If a transversal is perpendicular to two parallel lines, all eight angles are congruent. 3-2-2 Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. 3-2-3 Alternate Exterior Angles Theorem If two parallel lines are cut by a transversal, then the two pairs of alternate exterior angles are congruent. 3-2-4 Same-Side Interior Angles Theorem If two parallel lines are cut by a transversal, then the two pairs of same-side interior angles are supplementary. ∠1 ≅ ∠3 ∠2 ≅ ∠4 ∠5 ≅ ∠7 ∠

6 ≅ ∠8 m∠1 + m∠4 = 180° m∠2 + m∠3 = 180° You will prove Theorems 3-2-3 and 3-2-4 in Exercises 25 and 26. PROOF PROOF Alternate Interior Angles Theorem Given: ℓ ǁ m Prove: ∠ 2 ≅ ∠3 Proof: E X A M P L E 2 Finding Angle Measures Find each angle measure. A m∠EDF x = 125 m∠EDF = 125° Alt. Ext.  Thm. B m∠TUS 13x° + 23x° = 180° Same-Side Int.  Thm. 36x = 180 x = 5 m∠TUS = 23 (5) = 115° Combine like terms. Divide both sides by 36. Substitute 5 for x. 2. Find m∠ABD. 156 156 Chapter 3 Parallel and Perpendicular Lines �� Parallel Lines and Transversals When I solve problems with parallel lines and transversals, I remind myself that every pair of angles is either congruent or supplementary. If r ǁ s, all the acute angles are congruent and all the obtuse angles are congruent. The acute angles are supplementary to the obtuse angles. Nancy Martin East Branch High School E X A M P L E 3 Music Application The treble strings of a grand piano are parallel. Viewed from above, the bass strings form transversals to the treble strings. Find x and y in the diagram. By the Alternate Exterior Angles Theorem, (25x + 5y) ° = 125°. By the Corresponding Angles Postulate, (25x + 4y) ° = 120°. 25x + 5y = 125 - (25x + 4y = 120) ̶̶̶̶̶̶̶̶̶̶̶̶ y = 5 25x + 5 (5) = 125 x = 4, y = 5 Subtract the second equation from the first equation. Substitute 5 for y in 25x + 5y = 125. Simplify and solve for x. 3. Find the measures of the acute angles in the diagram.

THINK AND DISCUSS 1. Explain why a transversal that is perpendicular to two parallel lines forms eight congruent angles. 2. GET ORGANIZED Copy the diagram and graphic organizer. Complete the graphic organizer by explaining why each of the three theorems is true. 3- 2 Angles Formed by Parallel Lines and Transversals 157 157 �� 3-2 Exercises Exercises GUIDED PRACTICE Find each angle measure. p. 155 1. m∠JKL 2. m∠BEF KEYWORD: MG7 3-2 KEYWORD: MG7 Parent. m∠1 4. m∠CBY p. 156. Safety The railing of p. 157 a wheelchair ramp is parallel to the ramp. Find x and y in the diagram. Independent Practice For See Exercises Example 6–7 8–11 12 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S8 Application Practice p. S30 PRACTICE AND PROBLEM SOLVING Find each angle measure. 6. m∠KLM 7. m∠VYX 8. m∠ ABC 9. m∠EFG 10. m∠PQR 11. m∠STU 158 158 Chapter 3 Parallel and Perpendicular Lines �� 12. Parking In the parking lot shown, the lines that mark the width of each space are parallel. m∠1 = (2x - 3y) ° m∠2 = (x + 3y) ° Find x and y. Find each angle measure. Justify each answer with a postulate or theorem. 13. m∠1 14. m∠2 15. m∠3 16. m∠4 17. m∠5 18. m∠6 19. m∠7 Algebra State the theorem or postulate that is related to the measures of the angles in each pair. Then find the angle measures. 20. m∠1 = (7x + 15) °, m

∠2 = (10x - 9) ° 21. m∠3 = (23x + 11) °, m∠4 = (14x + 21) ° 22. m∠4 = (37x - 15) °, m∠5 = (44x - 29) ° 23. m∠1 = (6x + 24) °, m∠4 = (17x - 9) ° 24. Architecture The Luxor Hotel in Las Vegas, Nevada, is a 30-story pyramid. The hotel uses an elevator called an inclinator to take people up the side of the pyramid. The inclinator travels at a 39° angle. Which theorem or postulate best illustrates the angles formed by the path of the inclinator and each parallel floor? (Hint: Draw a picture.) 25. Complete the two-column proof of the Alternate Exterior Angles Theorem. Given: ℓ ǁ m Prove: ∠1 ≅ ∠2 Proof: Statements Reasons Architecture Architecture The Luxor hotel is 600 feet wide, 600 feet long, and 350 feet high. The atrium in the hotel measures 29 million cubic feet. 1. ℓ ǁ m 2. a.? ̶̶̶̶̶ 3. ∠3 ≅ ∠2 4. c.? ̶̶̶̶̶ 1. Given 2. Vert.  Thm. 3. b. 4. d.? ̶̶̶̶̶? ̶̶̶̶̶ 26. Write a paragraph proof of the Same-Side Interior Angles Theorem. Given: r ǁ s Prove: m∠1 + m∠2 = 180° Draw the given situation or tell why it is impossible. 27. Two parallel lines are intersected by a transversal so that the corresponding angles are supplementary. 28. Two parallel lines are intersected by a transversal so that the same-side interior angles are complementary. 3- 2 Angles Formed by Parallel Lines and Transversals 159 159 �� 29. This problem will prepare you for the Multi-Step TAKS Prep on page 180. In the diagram, which represents the side view of a mystery spot, m∠SRT = 25°.   RT is a transversal to  

PS and   QR. a. What type of angle pair is ∠QRT and ∠STR? b. Find m∠STR. Use a theorem or postulate to justify your answer. 30. Land Development A piece of property lies between two parallel streets as shown. m∠1 = (2x + 6) °, and m∠2 = (3x + 9) °. What is the relationship between the angles? What are their measures? 31. /////ERROR ANALYSIS///// In the figure, m∠ABC = (15x + 5) °, and m∠BCD = (10x + 25) °. Which value of m∠BCD is incorrect? Explain. 32. Critical Thinking In the diagram, ℓ ǁ m. Explain why x _ y = 1. 33. Write About It Suppose that lines ℓ and m are intersected by transversal p. One of the angles formed by ℓ and p is congruent to every angle formed by m and p. Draw a diagram showing lines ℓ, m, and p, mark any congruent angles that are formed, and explain what you know is true. 34. m∠RST = (x + 50) °, and m∠STU = (3x + 20) °. Find m∠RVT. 15° 27.5° 65° 77.5° 160 160 Chapter 3 Parallel and Perpendicular Lines �� 35. For two parallel lines and a transversal, m∠1 = 83°. For which pair of angle measures is the sum the least? ∠1 and a corresponding angle ∠1 and a same-side interior angle ∠1 and its supplement ∠1 and its complement 36. Short Response Given a ǁ b with transversal t, explain why �

�1 and ∠3 are supplementary. � � � � � � CHALLENGE AND EXTEND Multi-Step Find m∠1 in each diagram. (Hint: Draw a line parallel to the given parallel lines.) 37. �� 38. � �� 39. Find x and y in the diagram. Justify your answer. � 40. Two lines are parallel. The measures of two corresponding angles are a° and 2b°, and the measures of two same-side interior angles are a° and b°. Find the value of a. � �� SPIRAL REVIEW If the first quantity increases, tell whether the second quantity is likely to increase, decrease, or stay the same. (Previous course) 41. time in years and average cost of a new car 42. age of a student and length of time needed to read 500 words Use the Law of Syllogism to draw a conclusion from the given information. (Lesson 2-3) 43. If two angles form a linear pair, then they are supplementary. If two angles are supplementary, then their measures add to 180°. ∠1 and ∠2 form a linear pair. 44. If a figure is a square, then it is a rectangle. If a figure is a rectangle, then its sides are perpendicular. Figure ABCD is a square. Give an example of each angle pair. (Lesson 3-1) 45. alternate interior angles 46. alternate exterior angles 47. same-side interior angles � � � � � � � � 3- 2 Angles Formed by Parallel Lines and Transversals 161 161 3-3 Proving Lines Parallel TEKS G.3.C Geometric structure: use logical reasoning to prove statements are true.... Also G.1.A, G.3.E, G.9.A Objective Use the angles formed by a transversal to prove two lines are parallel. Who uses this? Rowers have to keep the oars on each side parallel in order to travel in a straight line. (See Example 4.) Recall that the converse of a theorem is found by exchanging the hypothesis and conclusion. The converse of a theorem is not automatically true. If it is true, it must be stated as a postulate or proved as a separate theorem. Postulate 3-3-1 Converse of the Corresponding Angles Postulate THEOREM HYPOTHESIS

CONCLUSION If two coplanar lines are cut by a transversal so that a pair of corresponding angles are congruent, then the two lines are parallel. ∠1 ≅ ∠ Using the Converse of the Corresponding Angles Postulate Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ ǁ m. A ∠1 ≅ ∠5 ∠1 ≅ ∠5 ℓ ǁ m ∠1 and ∠5 are corresponding angles. Conv. of Corr. ∠s Post. B m∠4 = (2x + 10) °, m∠8 = (3x - 55) °, x = 65 m∠4 = 2 (65) + 10 = 140 m∠8 = 3 (65) - 55 = 140 m∠4 = m∠8 ∠4 ≅ ∠8 ℓ ǁ m Substitute 65 for x. Substitute 65 for x. Trans. Prop. of Equality Def. of ≅  Conv. of Corr.  Post. Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ ǁ m. 1a. m∠1 = m∠3 1b. m∠7 = (4x + 25) °, m∠5 = (5x + 12) °, x = 13 162 162 Chapter 3 Parallel and Perpendicular Lines �� Postulate 3-3-2 Parallel Postulate Through a point P not on line ℓ, there is exactly one line parallel to ℓ. The Converse of the Corresponding Angles Postulate is used to construct parallel lines. The Parallel Postulate guarantees that for any line ℓ, you can always construct a parallel line through a point that is not on ℓ. Construction Parallel Lines  Draw a line ℓ and a point P that is not on ℓ.  Draw a line m through P that intersects ℓ. Label the angle 1.  Construct an angle congruent to ∠1 at P. By the converse of the Corresponding Angles Postulate, ℓ ǁ n. Theorems Proving Lines Parallel THEOREM HYPOTHESIS CONCLUSION

3-3-3 Converse of the Alternate ∠1 ≅ ∠2 Interior Angles Theorem If two coplanar lines are cut by a transversal so that a pair of alternate interior angles are congruent, then the two lines are parallel. 3-3-4 Converse of the Alternate ∠3 ≅ ∠4 Exterior Angles Theorem If two coplanar lines are cut by a transversal so that a pair of alternate exterior angles are congruent, then the two lines are parallel. 3-3-5 Converse of the Same-Side m∠5 + m∠6 = 180° Interior Angles Theorem If two coplanar lines are cut by a transversal so that a pair of same-side interior angles are supplementary, then the two lines are parallel You will prove Theorems 3-3-3 and 3-3-5 in Exercises 38–39. 3- 3 Proving Lines Parallel 163 163 �� PROOF PROOF Converse of the Alternate Exterior Angles Theorem Given: ∠1 ≅ ∠2 Prove: ℓ ǁ m Proof: It is given that ∠1 ≅ ∠2. Vertical angles are congruent, so ∠1 ≅ ∠3. By the Transitive Property of Congruence, ∠2 ≅ ∠3. So ℓ ǁ m by the Converse of the Corresponding Angles Postulate. E X A M P L E 2 Determining Whether Lines are Parallel Use the given information and the theorems you have learned to show that r ǁ s. A ∠2 ≅ ∠6 ∠2 ≅ ∠6 r ǁ s ∠2 and ∠6 are alternate interior angles. Conv. of Alt. Int.  Thm. B m∠6 = (6x + 18) °, m∠7 = (9x + 12) °, x = 10 m∠6 = 6x + 18 = 6 (10) + 18 = 78° Substitute 10 for x. m∠7 = 9x + 12 = 9 (10) + 12 = 102° m∠6 + m∠7 = 78° + 102° Substitute 10 for x. = 180° ∠6 and ∠7 are

same-side interior angles. r ǁ s Conv. of Same-Side Int.  Thm. Refer to the diagram above. Use the given information and the theorems you have learned to show that r ǁ s. 2a. m∠4 = m∠8 2b. m∠3 = 2x°, m∠7 = (x + 50) °, x = 50 E X A M P L E 3 Proving Lines Parallel Given: ℓ ǁ m, ∠1 ≅ ∠3 Prove: r ǁ p Proof: Statements Reasons 1. ℓ ǁ m 2. ∠1 ≅ ∠2 3. ∠1 ≅ ∠3 4. ∠2 ≅ ∠3 5. r ǁ p 1. Given 2. Corr.  Post. 3. Given 4. Trans. Prop. of ≅ 5. Conv. of Alt. Ext.  Thm. 3. Given: ∠1 ≅ ∠4, ∠3 and ∠4 are supplementary. Prove: ℓ ǁ m 164 164 Chapter 3 Parallel and Perpendicular Lines �� E X A M P L E 4 Sports Application During a race, all members of a rowing team should keep the oars parallel on each side. If m∠1 = (3x + 13) °, m∠2 = (5x - 5) °, and x = 9, show that the oars are parallel. A line through the center of the boat forms a transversal to the two oars on each side of the boat. ∠1 and ∠2 are corresponding angles. If ∠1 ≅ ∠2, then the oars are parallel. Substitute 9 for x in each expression: m∠1 = 3x + 13 = 3 (9) + 13 = 40° Substitute 9 for x in each expression. m∠2 = 5x - 5 = 5 (9) - 5 = 40° m∠1 = m∠2, so ∠1 ≅ ∠2. The corresponding angles are congruent, so the oars are parallel by the Converse of the Corresponding Angles Postulate. 4. What if…? Suppose the corresponding angles on the opposite side of

the boat measure (4y - 2) ° and (3y + 6) °, where y = 8. Show that the oars are parallel. THINK AND DISCUSS 1. Explain three ways of proving that two lines are parallel. 2. If you know m∠1, how could you use the measures of ∠5, ∠6, ∠7, or ∠8 to prove m ǁ n? 3. GET ORGANIZED Copy and complete the graphic organizer. Use it to compare the Corresponding Angles Postulate with the Converse of the Corresponding Angles Postulate. 3- 3 Proving Lines Parallel 165 165 21�� KEYWORD: MG7 3-3 KEYWORD: MG7 Parent 3-3 Exercises Exercises GUIDED PRACTICE. 162 Use the Converse of the Corresponding Angles Postulate and the given information to show that p ǁ q. 1. ∠4 ≅ ∠5 2. m∠1 = (4x + 16) °, m∠8 = (5x - 12) °, x = 28 3. m∠4 = (6x - 19) °, m∠5 = (3x + 14) °, x = 11 Use the theorems and given information to show that r ǁ s. p. 164 4. ∠1 ≅ ∠5 5. m∠3 + m∠4= 180° 6. ∠3 ≅ ∠7 7. m∠4 = (13x - 4) °, m∠8 = (9x + 16) °, x = 5 8. m∠8 = (17x + 37) °, m∠7 = (9x - 13) °, x = 6 9. m∠2 = (25x + 7) °, m∠6 = (24x + 12) °, 10. Complete the following two-column proof. p. 164 Given: ∠1 ≅ ∠2, ∠3 ≅ ∠1 Prove: XY ǁ WV Proof: Statements Reasons 1. ∠1 ≅ ∠2, ∠3 ≅ ∠1 1. Given 2. ∠2 ≅

∠3 3. b.? ̶̶̶̶̶ 2. a. 3. c.? ̶̶̶̶̶? ̶̶̶̶̶ 11. Architecture In the fire escape, p. 165 m∠1 = (17x + 9) °, m∠2 = (14x + 18) °, and x = 3. Show that the two landings are parallel. PRACTICE AND PROBLEM SOLVING Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ ǁ m. 12. ∠3 ≅ 7 13. m∠4 = 54°, m∠8 = (7x + 5) °, x = 7 14. m∠2 = (8x + 4) °, m∠6 = (11x - 41) °, x = 15 15. m∠1 = (3x + 19) °, m∠5 = (4x + 7) °, x = 12 166 166 Chapter 3 Parallel and Perpendicular Lines �� Independent Practice Use the theorems and given information to show that n ǁ p. For See Exercises Example 12–15 16–21 22 23 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S8 Application Practice p. S30 16. ∠3 ≅ ∠6 17. ∠2 ≅ ∠7 18. m∠4 + m∠6 = 180° 19. m∠1 = (8x - 7) °, m∠8 = (6x + 21) °, x = 14 20. m∠4 = (4x + 3) °, m∠5 = (5x -22) °, x = 25 21. m∠3 = (2x + 15) °, m∠5 = (3x + 15) °, x = 30 22. Complete the following two-column proof. Given: Prove: ̶̶ AB ǁ ̶̶ BC ǁ ̶̶ CD, ∠1 ≅ ∠2, ∠3 ≅ ∠4 ̶̶ DE Proof: Statements Reasons ̶̶ AB ǁ ̶̶ CD 1. 2. ∠1 ≅ ∠

3 3. ∠1 ≅ ∠2, ∠3 ≅ ∠4 4. ∠2 ≅ ∠4 5. d.? ̶̶̶̶̶̶ 1. Given 2. a. 3. b. 4. c. 5. e.? ̶̶̶̶̶̶? ̶̶̶̶̶̶? ̶̶̶̶̶̶? ̶̶̶̶̶̶ 23. Art Edmund Dulac used perspective when drawing the floor titles in this illustration for The Wind’s Tale by Hans Christian Andersen. Show that DJ ǁ EK if m∠1 = (3x + 2) °, m∠2 = (5x - 10) °, and x = 6. � � � � � � � � � � � � Name the postulate or theorem that proves that ℓ ǁ m. 24. ∠8 ≅ ∠6 25. ∠8 ≅ ∠4 26. ∠2 ≅ ∠6 27. ∠7 ≅ ∠5 28. ∠3 ≅ ∠7 29. m∠2 + m∠3 = 180° For the given information, tell which pair of lines must be parallel. Name the postulate or theorem that supports your answer. 30. m∠2 = m∠10 31. m∠8 + m∠9 = 180° 32. ∠1 ≅ ∠7 33. m∠10 = m∠6 34. ∠11 ≅ ∠5 35. m∠2 + m∠5 = 180° 36. Multi-Step Two lines are intersected by a transversal so that ∠1 and ∠2 are corresponding angles, ∠1 and ∠3 are alternate exterior angles, and ∠3 and ∠4 are corresponding angles. If ∠2 ≅ ∠4, what theorem or postulate can be used to prove the lines parallel? 3- 3 Proving Lines Parallel 167 167 �� 37. This problem will prepare you for the Multi-Step TAKS Prep on page 180. In the diagram, which represents the side view of a mystery spot, m∠SRT = 25°, and m∠SUR = 65°. a. Name a same-side interior angle of ∠S

UR ̶̶ for lines   SU and   RT with transversal RU. What is its measure? Explain your reasoning. b. Prove that   SU and   RT are parallel. 38. Complete the flowchart proof of the Converse of the Alternate Interior Angles Theorem. Given: ∠2 ≅ ∠3 Prove: ℓ ǁ m Proof: 39. Use the diagram to write a paragraph proof of the Converse of the Same-Side Interior Angles Theorem. Given: ∠1 and ∠2 are supplementary. Prove: ℓ ǁ m 40. Carpentry A plumb bob is a weight hung at the end of a string, called a plumb line. The weight pulls the string down so that the plumb line is perfectly vertical. Suppose that the angle formed by the wall and the roof is 123° and the angle formed by the plumb line and the roof is 123°. How does this show that the wall is perfectly vertical? 41. Critical Thinking Are the Reflexive, Symmetric, and Transitive Properties true for parallel lines? Explain why or why not. Reflexive: ℓ ǁ ℓ Symmetric: If ℓ ǁ m, then m ǁ ℓ. Transitive: If ℓ ǁ m and m ǁ n, then ℓ ǁ n. 42. Write About It Does the information given in the diagram allow you to conclude that a ǁ b? Explain. 43. Which postulate or theorem can be used to prove ℓ ǁ m? Converse of the Corresponding Angles Postulate Converse of the Alternate Interior Angles Theorem Converse of the Alternate Exterior Angles Theorem Converse of the Same-Side Interior Angles Theorem 168 168 Chapter 3 Parallel and Perpendicular Lines ��PlumblineWallRoof123˚123˚�� 44. Two coplanar lines are cut by a transversal. Which condition does NOT guarantee that the two lines are parallel? A pair of alternate interior angles are congruent. A pair of same-side interior

angles are supplementary. A pair of corresponding angles are congruent. A pair of alternate exterior angles are complementary. 45. Gridded Response Find the value of x so that ℓ ǁ m. �� CHALLENGE AND EXTEND Determine which lines, if any, can be proven parallel using the given information. Justify your answers. 46. ∠1 ≅ ∠15 48. ∠3 ≅ ∠7 50. ∠6 ≅ ∠8 47. ∠8 ≅ ∠14 49. ∠8 ≅ ∠10 51. ∠13 ≅ ∠11 52. m∠12 + m∠15 = 180° 53. m∠5 + m∠8 = 180° 54. Write a paragraph ̶̶ BD. proof that ̶̶ AE ǁ � � � �� Use the diagram for Exercises 55 and 56. 55. Given: m∠2 + m∠3 = 180° Prove: ℓ ǁ m 56. Given: m∠2 + m∠5 = 180° Prove: ℓ ǁ n � � � � � � � � � � � � � � � � �� SPIRAL REVIEW Solve each equation for the indicated variable. (Previous course) 57. a - b = -c, for a 58. y = 1 _ 2 x - 10, for x 59. 4y + 6x = 12, for y Write the converse, inverse, and contrapositive of each conditional statement. Find the truth value of each. (Lesson 2-2) 60. If an animal is a bat, then it has wings. 61. If a polygon is a triangle, then it has exactly three sides. 62. If the digit in the ones place of a whole number is 2, then the number is even. Identify each of the following. (Lesson 3-1) � 63. one pair of parallel segments 64. one pair of skew segments 65. one pair of perpendicular segments � � � � 3- 3 Proving Lines Parallel 169 169 3-3 Construct Parallel Lines In Lesson 3-3, you learned one method of constructing parallel lines using a compass and straightedge. Another method, called

the rhombus method, uses a property of a figure called a rhombus, which you will study in Chapter 6. The rhombus method is shown below. Use with Lesson 3-3 TEKS G.2.A Geometric structure: use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. Also G.9.A Activity 1 1 Draw a line ℓ and a point P not on the line. 2 Choose a point Q on the line. Place your compass point at Q and draw an arc through P that intersects ℓ. Label the intersection R. 3 Using the same compass setting as the 4 Draw PS ǁ ℓ. first arc, draw two more arcs: one from P, the other from R. Label the intersection of the two arcs S. Try This 1. Repeat Activity 1 using a different point not on the line. Are your results the same? 2. Using the lines you constructed in Problem 1, draw transversal PQ. Verify that the lines are parallel by using a protractor to measure alternate interior angles. 3. What postulate ensures that this construction is always possible? 4. A rhombus is a quadrilateral with four congruent sides. Explain why this method is called the rhombus method. 170 170 Chapter 3 Parallel and Perpendicular Lines �� Activity 2 1 Draw a line ℓ and point P on a piece of 2 Fold the paper through P so that both sides patty paper. of line ℓ match up 3 Crease the paper to form line m. P should 4 Fold the paper again through P so that be on line m. both sides of line m match up. 5 Crease the paper to form line n. Line n is parallel to line ℓ through P. Try This 5. Repeat Activity 2 using a point in a different place not on the line. Are your results the same? 6. Use a protractor to measure corresponding angles. How can you tell that the lines are parallel? 7. Draw a triangle and construct a line parallel to one side through the vertex that is not on that side. 8. Line m is perpendicular to both ℓ and n. Use this statement to complete the following conjecture: If two lines in a plane are perpendicular to the same line, then?. ̶̶̶̶̶̶

̶̶̶̶ 3- 3 Geometry Lab 171 171 3-4 Perpendicular Lines TEKS G.1.A Geometric structure:... connecting definitions... logical reasoning, and theorems. Also G.2.A, G.3.C, G.3.E, G.9.A Objective Prove and apply theorems about perpendicular lines. Vocabulary perpendicular bisector distance from a point to a line Why learn this? Rip currents are strong currents that flow away from the shoreline and are perpendicular to it. A swimmer who gets caught in a rip current can get swept far out to sea. (See Example 3.) The perpendicular bisector of a segment is a line perpendicular to a segment at the segment’s midpoint. A construction of a perpendicular bisector is shown below. Construction Perpendicular Bisector of a Segment  ̶̶ AB. Open the compass Draw wider than half of AB and draw an arc centered at A.  Using the same compass setting, draw an arc centered at B that intersects the first arc at C and D.  Draw   CD.   CD is the perpendicular bisector of ̶̶ AB. The shortest segment from a point to a line is perpendicular to the line. This fact is used to define the distance from a point to a line as the length of the perpendicular segment from the point to the line. E X A M P L E 1 Distance From a Point to a Line A Name the shortest segment from P to    AC. The shortest distance from a point to a line is the length of the perpendicular segment, ̶̶ PB is the shortest segment from P to   AC. so B Write and solve an inequality for x. ̶̶ PB is the shortest segment. Substitute x + 3 for PA and 5 for PB. Subtract 3 from both sides of the inequality. PA > PB x + 3 > 5 - 3 - 3 ̶̶̶ ̶̶̶̶ x > 2 1a. Name the shortest segment from A to   BC. 1b. Write and solve an inequality for x. 172 172 Chapter 3 Parallel and Perpendicular Lines �� Theorems THEOREM DIAGRAM EXAMPLE 3

-4-1 If two intersecting lines form a linear pair of congruent angles, then the lines are perpendicular. (2 intersecting lines form lin. pair of ≅  → lines ⊥.) 3-4-2 Perpendicular Transversal Theorem In a plane, if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other line. If two coplanar lines are perpendicular to the same line, then the two lines are parallel to each other. (2 lines ⊥ to same line → 2 lines ǁ.) 3-4- You will prove Theorems 3-4-1 and 3-4-3 in Exercises 37 and 38. PROOF PROOF Perpendicular Transversal Theorem Given: BC ǁ DE, AB ⊥ BC Prove: AB ⊥ DE Proof: It is given that BC ǁ DE, so ∠ABC ≅ ∠BDE by the Corresponding Angles Postulate. It is also given that   AB ⊥   BC, so m∠ABC = 90°. By the definition of congruent angles, m∠ABC = m∠BDE, so m∠BDE = 90° by the Transitive Property of Equality. By the definition of perpendicular lines,   AB ⊥   DE. E X A M P L E 2 Proving Properties of Lines Write a two-column proof. Given:   AD ǁ   BC,   AD ⊥   AB,   BC ⊥   DC Prove:   AB ǁ  �

�� DC Proof: Statements Reasons 1.   AD ǁ   BC,   BC ⊥   DC 1. Given 2.   AD ⊥   DC 3.   AD ⊥   AB 4.   AB ǁ   DC 2. ⊥ Transv. Thm. 3. Given 4. 2 lines ⊥ to same line → 2 lines ǁ. 2. Write a two-column proof. Given: ∠EHF ≅ ∠HFG,   FG ⊥   GH Prove:   EH ⊥   GH 3- 4 Perpendicular Lines 173 173 �� E X A M P L E 3 Oceanography Application Oceanography The National Weather Service in Brownsville provides a rip current forecast for South Padre Island and other Texas beaches. Park officials also post flags on the beach to warn of rip current danger. Source: www.ripcurrents. noaa.gov Rip currents may be caused by a sandbar parallel to the shoreline. Waves cause a buildup of water between the sandbar and the shoreline. When this water breaks through the sandbar, it flows out in a direction perpendicular to the sandbar. Why must the rip current be perpendicular to the shoreline? The rip current forms a transversal to the shoreline and the sandbar. The shoreline and the sandbar are parallel, and the rip current is perpendicular to the sandbar. So by the Perpendicular Transversal Theorem, the rip current is perpendicular to the shoreline. 3. A swimmer who gets caught in a rip current should swim in a direction perpendicular to the current. Why should the path of the swimmer be parallel to the shoreline? THINK AND DISCUSS 1. Describe what happens if two intersecting lines form a linear pair of congruent angles. 2. Explain why a transversal

that is perpendicular to two parallel lines forms eight congruent angles. 3. GET ORGANIZED Copy and complete the graphic organizer. Use the diagram and the theorems from this lesson to complete the table. 174 174 Chapter 3 Parallel and Perpendicular Lines Rip currentSandbarSandbarShoreline�� 3-4 Exercises Exercises KEYWORD: MG7 3-4 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary   CD is the perpendicular bisector of ̶̶ AB.   CD intersects What can you say about ̶̶ AB and   CD? What can you say about ̶̶ AC and ̶̶ AB at C. ̶̶ BC?. Name the shortest segment from p. 172 point E to   AD. 3. Write and solve an inequality for x. Complete the two-column proof. p. 173 Given: ∠ABC ≅ ∠CBE,   DE ⊥   AF Prove:   CB ǁ   DE Proof: Statements Reasons 1. ∠ABC ≅ ∠CBE 1. Given 2.   CB ⊥   AF 3. b.? ̶̶̶̶̶̶ 4.   CB ǁ   DE 2. a.? ̶̶̶̶̶̶ 3. Given 4. c.? ̶̶̶̶̶. 174 5. Sports The center line in a tennis court is perpendicular to both service lines. Explain why the service lines must be parallel to each other. Independent Practice For See Exercises Example 6–7 8 9 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S8 Application Practice p. S30 PRACTICE AND PROBLEM SOLVING 6. Name the shortest segment from point W to ̶̶ XZ. 7. Write and

solve an inequality for x. 8. Complete the two-column proof below. Given:   AB ⊥   BC, m∠1 + m∠2 = 180° Prove:   BC ⊥   CD Proof: Statements Reasons 1.   AB ⊥   BC 2. m∠1 + m∠2 = 180° 1. Given 2. a.? ̶̶̶̶̶ 3. ∠1 and ∠2 are supplementary. 3. Def. of supplementary 4. b.? ̶̶̶̶̶ 5.   BC ⊥   CD 4. Converse of the Same-Side Interior Angles Theorem 5. c.? ̶̶̶̶̶ 3- 4 Perpendicular Lines 175 175 ��ServicelineCenterlineServicelinege07se_c03l04003ad2ndpass3/18/05NPatel�� 9. Music The frets on a guitar are all perpendicular to one of the strings. Explain why the frets must be parallel to each other. For each diagram, write and solve an inequality for x. 10. 11. Multi-Step Solve to find x and y in each diagram. 12. 14. 13. �� 15. Determine if there is enough information given in the diagram to prove each statement. 16. ∠1 ≅ ∠2 17. ∠1 ≅ ∠3 18. ∠2 ≅ ∠3 19. ∠2 ≅ ∠4 20. ∠3 ≅ ∠4 21. ∠3 ≅ ∠5 22. Critical Thinking Are the Reflexive, Symmetric, and Transitive Properties true for perpendicular lines? Explain why or why not. Reflexive: ℓ ⊥ ℓ Symmetric: If ℓ ⊥ m, then m ⊥ ℓ. Transitive: If ℓ ⊥ m and m ⊥ n, then ℓ ⊥ n. 23

. This problem will prepare you for the Multi-Step TAKS Prep on page 180. a mystery spot, In the diagram, which represents the side view of ̶̶ PQ, ̶̶ PS ⊥ ̶̶ PQ ǁ ̶̶ RS. ̶̶ RS, and a. Prove ̶̶ PS ǁ ̶̶ QR. ̶̶ QR ⊥ ̶̶ RS and ̶̶ PS. ̶̶ QR ⊥ ̶̶ PQ ⊥ b. Prove 176 176 Chapter 3 Parallel and Perpendicular Lines �� 24. Geography Felton Avenue, Arlee Avenue, and Viehl Avenue are all parallel. Broadway Street is perpendicular to Felton Avenue. Use the satellite photo and the given information to determine the values of x and y. 25. Estimation Copy the diagram onto a grid with 1 cm by 1 cm squares. Estimate the distance from point P to line ℓ. �� 26. Critical Thinking Draw a figure to show that Theorem 3-4-3 is not true if the lines are not in the same plane. 27. Draw a figure in which perpendicular bisector of ̶̶ AB. ̶̶ AB is a perpendicular bisector of ̶̶ XY but ̶̶ XY is not a 28. Write About It A ladder is formed by rungs that are perpendicular to the sides of the ladder. Explain why the rungs of the ladder are parallel. �� Construction Construct a segment congruent to each given segment and then �� construct its perpendicular bisector. 29. 30. 31. Which inequality is correct for the given diagram? 2x + 5 < 3x x > 1 2x + 5 > 3x x > 5 32. In the diagram, ℓ ⊥ m. Find x and y. x = 5, y = 7 x = 7, y = 5 x = 90, y = 90 x = 10, y = 5 33. If ℓ ⊥ m, which statement is NOT correct? m∠2 = 90° m∠1 + m∠2 = 180° ∠1 ≅ ∠2 ∠1 �

�� ∠2 3- 4 Perpendicular Lines 177 177 �� 34. In a plane, both lines m and n are perpendicular to both lines p and q. Which conclusion CANNOT be made All angles formed by lines m, n, p, and q are congruent. 35. Extended Response Lines m and n are parallel. Line p intersects line m at A and line n at B, and is perpendicular to line m. a. What is the relationship between line n and line p? Draw a diagram to support your answer. b. What is the distance from point A to line n? What is the distance from point B to line m? Explain. c. How would you define the distance between two parallel lines in a plane? CHALLENGE AND EXTEND 36. Multi-Step Find m∠1 in the diagram. (Hint: Draw a line parallel to the given parallel lines.) 37. Prove Theorem 3-4-1: If two intersecting lines form a linear pair of congruent angles, then the two lines are perpendicular. 38. Prove Theorem 3-4-3: If two coplanar lines are perpendicular to the same line, then the two lines are parallel to each other. � � � � � � SPIRAL REVIEW 39. A soccer league has 6 teams. During one season, each team plays each of the other teams 2 times. What is the total number of games played in the league during one season? (Previous course) Find the measure of each angle. (Lesson 1-4) 40. the supplement of ∠DJE 41. the complement of ∠FJG 42. the supplement of ∠GJH For the given information, name the postulate or theorem that proves ℓ ǁ m. (Lesson 3-3) 43. ∠2 ≅ ∠7 44. ∠3 ≅ ∠6 45. m∠4 + m∠6 = 180° � � �� 178 178 Chapter 3 Parallel and Perpendicular Lines 3-4 Use with Lesson 3-4 Activity Construct Perpendicular Lines In Lesson 3-4, you learned to construct the perpendicular bisector of a segment. This is the basis of the construction of a line perpendicular to a given line through a given point. The

steps in the construction are the same whether the point is on or off the line. TEKS G.2.A Geometric structure: use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. Also G.9.A Copy the given line ℓ and point P. 1 Place the compass point on P and draw an arc that intersects ℓ at two points. Label the points A and B. 2 Construct the perpendicular bisector of ̶̶ AB. Try This Copy each diagram and construct a line perpendicular to line ℓ through point P. Use a protractor to verify that the lines are perpendicular. 1. 2. 3. Follow the steps below to construct two parallel lines. Explain why ℓ ǁ n. Step 1 Given a line ℓ, draw a point P not on ℓ. Step 2 Construct line m perpendicular to ℓ through P. Step 3 Construct line n perpendicular to m through P. 3- 4 Geometry Lab 179 179 �� SECTION 3A Parallel and Perpendicular Lines and Transversals On the Spot Inside a mystery spot building, objects can appear to roll uphill, and people can look as if they are standing at impossible angles. This is because there is no view of the outside, so the room appears to be normal. Suppose that the ground is perfectly level and the floor of the building forms a 25° angle with the ground. The floor and ceiling are parallel, and the walls are perpendicular to the floor. 1. A table is placed in the room. The legs of the table are perpendicular to the floor, and the top is perpendicular to the legs. Draw a diagram and describe the relationship of the tabletop to the floor, walls, and ceiling of the room. 2. Find the angle of the table top relative to the ground. Suppose a ball is placed on the table. Describe what would happen and how it would appear to a person in the room. 3. Two people of the same height are standing on opposite ends of a board that makes a 25° angle with the floor, as shown. Explain how you know that the board is parallel to the ground. What would appear to be happening from the point of view of a person inside the room? 4. In the room, a lamp hangs from the ceiling along a line perpendicular to the ground. Find the angle the line makes with the walls. Describe how it would appear to a person standing in the room.

180 180 Chapter 3 Parallel and Perpendicular Lines �� Quiz for Lessons 3-1 Through 3-4 SECTION 3A 3-1 Lines and Angles Identify each of the following. 1. a pair of perpendicular segments 2. a pair of skew segments 3. a pair of parallel segments 4. a pair of parallel planes Give an example of each angle pair. 5. alternate interior angles 6. alternate exterior angles 7. corresponding angles 8. same-side interior angles 3-2 Angles Formed by Parallel Lines and Transversals Find each angle measure. 9. 10. 11. 3-3 Proving Lines Parallel Use the given information and the theorems and postulates you have learned to show that a ǁ b. 12. m∠8 = (13x + 20) °, m∠6 = (7x + 38) °, x = 3 13. ∠1 ≅ ∠5 14. m∠8 + m∠7 = 180° 15. m∠8 = m∠4 16. The tower shown is supported by guy wires such that m∠1 = (3x + 12) °, m∠2 = (4x - 2) °, and x = 14. Show that the guy wires are parallel. 3-4 Perpendicular Lines 17. Write a two-column proof. Given: ∠1 ≅ ∠2, ℓ ⊥ n Prove: ℓ ⊥ p Ready to Go On? 181 181 �� 3-5 Slopes of Lines TEKS G.7.B Dimensionality and the geometry of location: use slopes... to investigate geometric relationships, including parallel lines, perpendicular lines,.... Also G.7.A, G.7.C Objectives Find the slope of a line. Use slopes to identify parallel and perpendicular lines. Vocabulary rise run slope Why learn this? You can use the graph of a line to describe your rate of change, or speed, when traveling. (See Example 2.) The slope of a line in a coordinate plane is a number that describes the steepness of the line. Any two points on a line can be used to determine the slope. Slope of a Line DEFINITION EXAMPLE The rise is the difference in the y-values of two points on

a line. The run is the difference in the x-values of two points on a line. The slope of a line is the ratio of rise to run. If ( x 1, y 1 ) and ( x 2, y 2 ) are any two points on line, the slope of the line is Finding the Slope of a Line Use the slope formula to determine the slope of each line. A    AB B    CD A fraction with zero in the denominator is undefined because it is impossible to divide by zero. Substitute (2, 3) for ( x 1, y 1 ) and (7, 5) for ( x 2, y 2 ) in the slope formula and then simplify Substitute (4, -3) for ( x 1, y 1 ) and (4, 5) for ( x 2, y 2 ) in the slope formula and then simplify. 5 - (-3 The slope is undefined. = 8 _ 0 m = 182 182 Chapter 3 Parallel and Perpendicular Lines �� Use the slope formula to determine the slope of each line. C   EF D  GH Substitute (3, 4) for ( x 1, y 1 ) and (6, 4) for ( x 2, y 2 ) in the slope formula and then simplify0 Substitute (6, 2) for ( x 1, y 1 ) and (2, 6) for ( x 2, y 2 ) in the slope formula and then simplify4 = -1 1. Use the slope formula to determine the slope of   JK through J (3, 1) and K (2, -1). Positive Slope Negative Slope Zero Slope Undefined Slope Summary: Slope of a Line One interpretation of slope is a rate of change. If y represents miles traveled and x represents time in hours, the slope gives the rate of change in miles per hour. E X A M P L E 2 Transportation Application Transportation A trip from Dallas to Atlanta, Georgia covers 781 miles and crosses three states but is still shorter than a trip across Texas. The distance from Orange, Texas to El Paso via Interstate 10 is 859 miles. Tony is driving from Dallas, Texas,

to Atlanta, Georgia. At 3:00 P.M., he is 180 miles from Dallas. At 5:30 P.M., he is 330 miles from Dallas. Graph the line that represents Tony’s distance from Dallas at a given time. Find and interpret the slope of the line. Use the points (3, 180) and (5.5, 330) to graph the line and find the slope. m = 330 - 180 _ 5.5 - 3 = 150 _ 2.5 = 60 The slope is 60, which means he is traveling at an average speed of 60 miles per hour. 2. What if…? Use the graph above to estimate how far Tony will have traveled by 6:30 P.M. if his average speed stays the same. 3- 5 Slopes of Lines 183 183 �� Slopes of Parallel and Perpendicular Lines 3-5-1 Parallel Lines Theorem In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel. 3-5-2 Perpendicular Lines Theorem In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1. Vertical and horizontal lines are perpendicular. If a line has a slope of a_ b and - b _ a are called opposite reciprocals. The ratios a _ b, then the slope of a perpendicular line is - b_ a. E X A M P L E 3 Determining Whether Lines Are Parallel, Perpendicular, or Neither Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. A B C Four given points do not always determine two lines. Graph the lines to make sure the points are not collinear.    AB and    CD for A (2, 1), B (1, 5), C (4, 2), and D (5, -2) slope of   AB = 5 - 1 _ = -4 1 - 2 = 4 _ -1 = -4 _ 1 slope of   CD = -2 - 2 _ 5 - 4 The lines have the same slope, so they are

parallel. = -4    ST and    UV for S (-2, 2), T (5, -1), U (3, 4), and V (-1, -4) slope of   ST = -1 - 2 _ 5 - (-2) slope of   UV = -4 - 4 _ -1 - 3 = -8 _ -4 = 2 The slopes are not the same, so the lines are not parallel. The product of the slopes is not -1, so the lines are not perpendicular.    FG and    HJ for F (1, 1), G (2, 2), H (2, 1), and J (1, 2) slope of   FG = 2 - 1 _ = 1 slope of   HJ = 2 - 1 _ = -1 = 1 _ 1 = 1 _ -1 2 - 1 1 - 2 The product of the slopes is 1 (-1) = -1, so the lines are perpendicular. Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. 3a.   WX and   YZ for W (3, 1), X (3, -2), Y (-2, 3), and Z (4, 3) 3b.   KL and   MN for K (-4, 4), L (-2, -3), M (3, 1), and N (-5, -1) 3c.   BC and   DE for B (1, 1), C (3, 5), D (-2, -6), and E (3, 4) 184 184 Chapter 3 Parallel and Perpendicular Lines �� THINK AND DISCUSS 1. Explain how to find the slope of a line when given two points. 2. Compare the slopes of horizontal and vertical

lines. 3. GET ORGANIZED Copy and complete the graphic organizer. 3-5 Exercises Exercises KEYWORD: MG7 3-5 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary The slope of a line is the ratio of its? to its ̶̶̶?. (rise or run) ̶̶̶ Use the slope formula to determine the slope of each line. p. 182 2.  MN 3.   CD 4.  AB 5.   ST. 183 6. Biology A migrating bird flying at a constant speed travels 80 miles by 8:00 A.M. and 200 miles by 11:00 A.M. Graph the line that represents the bird’s distance traveled. Find and interpret the slope of the line. 184 Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. 7.   HJ and   KM for H (3, 2), J (4, 1), K (-2, -4), and M (-1, -5) 8.   LM and   NP for L (-2, 2), M (2, 5), N (0, 2), and P (3, -2) 9.   QR and   ST for Q (6, 1), R (-2, 4), S (5, 3), and T (-3, -1) 3- 5 Slopes of Lines 185 185 �� Independent Practice Use the slope formula to determine the slope of each line. PRACTICE AND PROBLEM SOLVING For See Exercises Example 10.  AB 11.   CD 10–13 14 15–17 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S9 Application Practice p. S30

12.  EF 13.   GH 14. Aviation A pilot traveling at a constant speed flies 100 miles by 2:30 P.M. and 475 miles by 5:00 P.M. Graph the line that represents the pilot’s distance flown. Find and interpret the slope of the line. Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. 15.  AB and   CD for A (2, -1), B (7, 2), C (2, -3), and D (-3, -6) 16.  XY and   ZW for X (-2, 5), Y (6, -2), Z (-3, 6), and W (4, 0) 17.  JK and   JL for J (-4, -2), K (4, -2), and L (-4, 6) 18. Geography The Rio Grande has an elevation of about 1150 meters above sea level in El Paso. The length of the river from that point to Brownsville where it enters the sea is about 2400 km. Find and interpret the slope of the river. For F (7, 6), G (-3, 5), H (-2, -3), J (4, -2), and K (6, 1), find each slope. 19.   FG 20.   GJ 21.  HK 22.   GK 23. Critical Thinking The slope of AB is greater than 0 and less than 1. Write an inequality for the slope of a line perpendicular to   AB. 24. Write About It Two cars are driving at the same speed. What is true about the lines that represent the distance traveled by each car at a given time? 25. This problem will prepare you for the Multi-Step TAKS Prep on page 200. A traffic engineer calculates the

speed of vehicles as they pass a traffic light. While the light is green, a taxi passes at a constant speed. After 2 s the taxi is 132 ft past the light. After 5 s it is 330 ft past the light. a. Find the speed of the taxi in feet per second. b. Use the fact that 22 ft/s = 15 mi/h to find the taxi’s speed in miles per hour. 186 186 Chapter 3 Parallel and Perpendicular Lines �� 26.   AB ⊥   CD for A (1, 3), B (4, -2), C (6, 1), and D (x, y). Which are possible values of x and y? x = 1, y = -2 x = 3, y = 6 x = 3, y = -4 x = -2, y = -4 27. Classify   MN and   PQ for M (-3, 1), N (1, 3), P (8, 4), and Q (2, 1). Parallel Perpendicular Vertical Skew 28. In the formula d = rt, d represents distance, and r represents the rate of change, or slope. Which ray on the graph represents a slope of 45 miles per hour? A B C D CHALLENGE AND EXTEND Use the given information to classify   29. a = c JK for J (a, b) and K (c, d). 30. b = d 31. The vertices of square ABCD are A (0, -2), B (6, 4), C (0, 10), D (-6, 4). a. Show that the opposite sides are parallel. b. Show that the consecutive sides are perpendicular. c. Show that all sides are congruent. 32.   ST ǁ   VW for S (-3, 5), T (1, -1), V (x, -3), and W (1, y). Find a set of possible values for x and y. 33. �