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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 conclusi... |
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 Dete... |
. 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... |
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 Ali... |
. 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 fr... |
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 ... |
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 work... |
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 wit... |
, 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 coll... |
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... |
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 =... |
. 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 releas... |
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 8... |
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... |
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 slee... |
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 finish... |
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... |
. 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... |
̶̶ 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 ≅ Refl... |
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 ... |
) 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 th... |
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 T... |
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 ̶... |
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 ≅ ̶... |
� 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.” ... |
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 pr... |
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 ... |
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 ... |
�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: ̶̶ ... |
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 pai... |
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. Us... |
≅ ∠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, po... |
��������� 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 de... |
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 Rea... |
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 r... |
= 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... |
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... |
�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 parag... |
= 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° + ... |
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. ... |
�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 ... |
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 f... |
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 �������... |
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 compoun... |
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 defi... |
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. ... |
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.” Fi... |
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 Satur... |
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.... |
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... |
�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 ... |
� ∠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 ���������������������... |
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 ... |
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 (... |
. 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 to... |
���������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������� KEYWORD: MG7 TestPre... |
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 ... |
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... |
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 tw... |
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 f... |
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.... |
★ ★ 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, includi... |
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 Co... |
� 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 segment... |
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 an... |
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 inter... |
��. 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 ... |
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 ... |
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 = ... |
✓ 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... |
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 Correspond... |
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° = 1... |
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 15... |
∠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 inc... |
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 measure... |
�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° a... |
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... |
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 th... |
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 R... |
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... |
∠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 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. � ... |
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 pro... |
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... |
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 Acti... |
̶̶̶̶ 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 t... |
-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.... |
�� 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 Perpendicula... |
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�����������������������������... |
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. C... |
. 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 ��������������... |
�� ∠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... |
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 ar... |
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 e... |
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 ... |
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... |
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... |
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. ... |
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, perpen... |
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... |
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