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/07 4:40 PM Page 44 44 Logic A compound sentence may contain both negations and conjunctions at the same time. For example: Let p represent “Ten is divisible by 2.” Let q represent “Ten is divisible by 3.” Then p ∧ q represents “Ten is divisible by 2 and ten is not divisible by 3.” Here p is true, q is false, and q is ... |
� q) F T Answers: a. (p ∧ q) b. True EXAMPLE 2 Solution Use the domain {1, 2, 3, 4} to find the truth set for the open sentence (x 3) ∧ (x is a prime) Let x 1 Let x 2 Let x 3 Let x 4 (1 3) ∧ (1 is a prime) (2 3) ∧ (2 is a prime) (3 3) ∧ (3 is a prime) (4 3) ∧ (4 is a prime) T ∧ F False T ∧ T True F ∧ T False F ∧ F Fals... |
Writing About Mathematics 1. Is the negation of a conjunction, (p ∧ q), the same as p ∧ q? Justify your answer. 2. What must be the truth values of p, q, and r in order for (p ∧ q) ∧ r to be true? Explain your answer. Developing Skills In 3–12, write each sentence in symbolic form, using the given symbols. Let p repre... |
is false, then p ∧ q is _______. 23. If p is true, or q is true, but not both, then p ∧ q is _______. 24. When p ∧ q is true, then p is ______ and q is ______. 25. When p ∧ q is true, then p is ______ and q is ______. 26. When p ∧ q is true, then p is ______ and q is ______. 27. When p is false and q is true, then (p ... |
that I a. In winter I wear a hat. b. In winter I wear a scarf. c. In winter I do not wear a hat. should. (True) a. I do not practice. b. I know that I should practice. c. I practice. 2-3 DISJUNCTIONS In logic, a disjunction is a compound statement formed by combining two simple statements using the word or. Each of th... |
disjunction, p ∨ q EXAMPLE 1 Use the following statements: Let k represent “Kurt plays baseball.” Let a represent “Alicia plays baseball.” Let n represent “Nathan plays soccer.” Write each given sentence in symbolic form. a. Kurt or Alicia play baseball. b. Kurt plays baseball or Nathan plays soccer. c. Alicia plays b... |
, 4, 5, 6, 7}. a. The solution set of x 4 is {1, 2, 3} and the solution set of x 3 is {4, 5, 6, 7}. The solution set of the disjunction (x 4) ∨ (x 3) includes all the numbers that make x 4 true together with all the numbers that make x 3 true. Answer {1, 2, 3, 4, 5, 6, 7} Note: The solution set of the disjunction (x 4)... |
this the exclusive or. The truth table for the exclusive or will be different from the table shown for disjunction. In the exclusive or, the disjunction p or qwill be true when p is true, or when q is true, but not both. In everyday conversation, it is often evident from the context which of these uses of or is intend... |
1,000 milligrams, and a gram is a measure of length. In 13–20, symbols are assigned to represent sentences. Let b represent “Breakfast is a meal.” Let s represent “Spring is a season.” Let h represent “Halloween is a season.” For each sentence given in symbolic form: a. Write a complete sentence in words to show what ... |
(True) Michelle is my friend. (?) 32. I practice the cello on Monday or I practice the piano on Monday. (True) I do not practice the piano on Monday. (False) I practice the cello on Monday. (?) 2-4 CONDITIONALS A sentence such as “If I have finished my homework, then I will go to the movies” is frequently used in dail... |
sometimes referred to as the consequent. It is the part of a sentence that closes an argument. The conclusion usually follows the word then. There are different ways to write the conditional. When the conditional uses the word if, the hypothesis always follows if. When the conditional uses the word implies, the hypoth... |
do if you did get an A. The conditional statement is true. p → q T T p q F CASE 4 You do not get an A in Geometry. (p is false.) I do not buy you a new graphing calculator. (q is false.) Since you did not get an A, I do not have to keep our agreement. The conditional statement is true. p F q F p → q T Case 2 tells us ... |
will learn. (2) The assignment will be completed if I work at it every day. i f q p a. p: I work at it every day. b. q: The assignment will be completed. (3) Hidden Conditional: If we all work together and do our best, then the task is easy. i p e q a. p: We all work together and we do our best. b. q: the task is easy... |
“There are 52 weeks in a year.” Let h represent “An hour has 75 minutes.” (True) (True) (False) (1) If Monday is the first day of the week, then there are 52 weeks in a year. (2) If there are 52 weeks in a year, then an hour has 75 minutes. (3) If there are not 52 weeks in a year then Monday is the first day of the we... |
tire. 11. If the car has a flat tire, then Danny will change the tire. 12. If Danny has a spare tire, then Danny will change the tire. 13. If the car does not have a flat tire, then Danny will not change the tire. 14. Danny will not change the tire if Danny doesn’t have a spare tire. 15. The car has a flat tire if Dan... |
flowers.” (True) (False) (True) (True) For each compound statement in symbolic form: a. Write a complete sentence in words to show what the symbols represent. b. Tell whether the compound statement is true or false. 32. j → g 36. (j ∧ f ) → d 33. d → g 37. (j ∧ g) → f 34. f → g 38. j → (d ∧ f ) 35. g → j 39. g → (j ∨ ... |
(True) Area (?) 1 2bh 2-5 INVERSES, CONVERSES, AND CONTRAPOSITIVES The conditional is the most frequently used statement in the construction of an argument or in the study of mathematics. We will use the conditional frequently in our study of geometry. In order to use the conditional statements correctly, we must unde... |
are both right angles.” Conditional (p → q): If two angles are congruent, g p then the two angles are both right angles. i q Inverse (p → q): If two angles are not congruent, i p then the two angles are not both right angles. i q We can find the truth value of these two statements with two angles A and B when mA 60 an... |
same truth value. The conditional (p → q) and its inverse (p → q) have opposite truth values when p and q have opposite truth values. Conditional Inverse 14365C02.pgs 7/9/07 4:40 PM Page 63 Inverses, Converses, and Contrapositives 63 The Converse The converse of a conditional statement is formed by interchanging the h... |
�Today is Friday.” Let q represent “Tomorrow is Saturday.” Conditional (p → q): If today is Friday, then tomorrow is Saturday. e f Converse (q → p): If tomorrow is Saturday, then today is Friday. f e q p p q 14365C02.pgs 7/9/07 4:40 PM Page 64 64 Logic On Friday, p is true and q is true. Therefore, both (p → q) and (q ... |
and (p → q) is true. Similarly, if p and q are false, p is true and q is true, so (q → p) is true. 14365C02.pgs 7/9/07 4:40 PM Page 65 Inverses, Converses, and Contrapositives 65 2. A false conditional can have a false contrapositive. Conditional If x is an odd number, then x is a prime number. f f p q Contrapositive ... |
) are contrapositives of each other. Since a conditional and its contrapositive always have the same truth value, the converse and the inverse always have the same truth value. This can be verified by constructing the following truth table EXAMPLE 1 Write the inverse, converse, and contrapositive of the given condition... |
the conditional is true. Do you agree with Samuel? Explain why or why not. 2. Kate said that if you know the truth value of a conditional and of its converse then you know the truth value of the inverse and the contrapositive. Do you agree with Kate? Explain why or why not. Developing Skills In 3–6, for each statement... |
measure, then the triangle is equiangular. 23. If 0, then 1 2 1 2 is a counting number. In 24–28, write the numeral preceding the expression that best answers the question. 24. When p → q is true, which related conditional must be true? (1) q → p (2) p → q (3) p → q (4) q → p 25. Which is the contrapositive of “If Mar... |
Applying Skills In 29–34, assume that each conditional statement is true. Then: a. Write its converse in words and state whether the converse is always true, sometimes true, or never true. b. Write its inverse in words and state whether the inverse is always true, sometimes true, or never true. c. Write its contraposi... |
Every definition is a true biconditional. Every definition can be written in reverse order. Both of the following statements are true: • Congruent segments are segments that have the same measure. • Line segments that have the same measure are congruent. We can restate the definition as two true conditionals: • If two... |
the true biconditional (p ∧ q) ↔ (p ∨ q). p ∧ q EXAMPLE 1 Determine the truth value to be assigned to the biconditional. Germany is a country in Europe if and only if Berlin is the capital of Germany. Solution “Germany is a country in Europe” is true. “Berlin is the capital of Germany” is true. Therefore, the bicondit... |
do not always have the same truth value. Therefore, the biconditional “Today is Monday if and only if I go to basketball practice” is not always true. We usually say that a statement that is not always true is false. Answer EXAMPLE 3 Determine the truth value of the biconditional. 3y 1 28 if and only if y 9. 14365C02.... |
x 5 if and only if x 3. 16. Today is Friday if and only if tomorrow is Saturday. 14365C02.pgs 7/9/07 4:40 PM Page 74 74 Logic Applying Skills 17. Let p represent “x is divisible by 2.” Let q represent “x is divisible by 3.” Let r represent “x is divisible by 6.” a. Write the biconditional (p ∧ q) ↔ r in words. b. Show... |
truth of related facts. To do this, we can look for patterns that are frequently used in drawing conclusions. These patterns are called the laws of logic. 14365C02.pgs 7/9/07 4:40 PM Page 75 The Laws of Logic 75 The Law of Detachment A valid argument uses a series of statements called premises that have known truth va... |
Disjunctive Inference We know that a disjunction is true when one or both statements that make up the disjunction are true. The disjunction is false when both statements that make up the disjunction are false. For example, let p represent “A real number is rational” and q represent “A real number is irrational.” Then ... |
” must be true. Alternative Solution Let p represent “I walk to school.” Let q represent “I ride to school with my friend.” 14365C02.pgs 7/9/07 4:40 PM Page 77 The Laws of Logic 77 Make a truth table for the disjunction p ∨ q and eliminate the rows that do not apply. We know that p ∨ q is true. We also know that since ... |
Eliminate the first row of truth values in which p is true. The second row in which p is true has already been eliminated. (3) Two rows remain, one in which q is true and the other in which q is false Answer “I will go to the library” could be either true or false. EXAMPLE 4 Draw a conclusion or conclusions base on th... |
I like swimming or kayaking. I like kayaking. I like swimming. 7. x 18 if x 14. x 14 x 18 I speed. I get a ticket. 6. I like swimming or kayaking. I do not like swimming. I like kayaking. 8. I live in Pennsylvania if I live in Philadelphia. I do not live in Philadelphia. I live in Pennsylvania. 9. If I am late for din... |
, we did not go bowling. I do not take woodshop. 25. Five is a prime if and only if five has 26. If x is an integer greater than 2 and x exactly two factors. Five is a prime. is a prime, then x is odd. x is not an odd integer. 27. If a ray bisects an angle, the ray divides the angle into two congruent angles. Ray DF do... |
so c is true. (Law of Detachment) Also, c is false. Using Statement 2 c ∨ b is true and c is false, so b must be true. (Law of Disjunctive Inference) Answer Rachel does not join the choir. Rachel will play basketball. EXAMPLE 2 If Alice goes through the looking glass, then she will see Tweedledee. If Alice sees Tweedl... |
take a different course in one of three areas for their senior year: mathematics, art, and thermodynamics. The following statements about the siblings are known to be true. • Ted tutors his sibling taking the mathematics course. • The art student and Ted have an argument over last night’s basketball game. • Mary loves... |
02.pgs 7/9/07 4:40 PM Page 84 84 Logic 7. When p → q and p ∧ q are both false, what are the truth values of p and of q? 8. When p → q is true and p ∧ q is false, what are the truth values of p and of q? 9. When p → q is true, p ∨ r is true and q is false, what is the truth value of r? Applying Skills 10. Laura, Marta, ... |
game in which each decides to be either a liar or a truthteller. A liar must always lie and a truthteller must always tell the truth. When you met these friends, you asked Augustus which he had chosen to be. You didn’t hear his answer but Brutus volunteered, “Augustus said that he is a liar.” Caesar added, “If one of ... |
called a consequent, is an ending or a sentence that closes an argument. The conclusion usually follows the word then. • Beginning with a statement (p → q), the inverse (p → q) is formed by negating the hypothesis and negating the conclusion. • Beginning with a statement (p → q), the converse (q → p) is formed by inte... |
q • Hypothesis • Premise • Antecedent • Conclusion • Consequent 2-5 Inverse • Converse • Contrapositive • Logical equivalents 2-6 Biconditional 2-7 Laws of logic • Valid argument • Premises • Law of Detachment • Law of Disjunctive Inference REVIEW EXERCISES 1. The statement “If I go to school, then I do not play baske... |
. 16. When p ∨ q is false, then p is _______ and q is _______. 17. If the conclusion q is true, then p → q must be _______. In 18–22, find the truth value of each sentence when a, b, and c are all true. 18. a 20. b → c 21. a ∨ b 19. b ∧ c 22. a ↔ b In 23–32, let p represent “x 5,” and let q represent “x is prime.” Use ... |
amended the conditions, by moving a comma, to read: I leave $100,000 to each of my nieces who, at the time of my death, is over 21, or unmarried and does not smoke. Which nieces described in Exercise 35 will now inherit $100,000? 37. At a swim meet, Janice, Kay, and Virginia were the first three finishers of a 200-met... |
two tautologies, one using symbols and the other using words. 2. A contradiction is a statement that is always false, that is, it cannot be true under any circumstances. For instance, the conjunction p ∧ p is a contradiction because if p is true, p is false and the conjunction is false, and if p is false, the conjunct... |
is Saturday or I have to go to school. (4) Today is not Saturday. 5. Which of the following is not a requirement in order for point H to be (3) GH HI GI (4) G, H, and I are distinct points. between points G and I? (1) GH HI (2) G, H, and I are collinear. h UW mTUV? (1) 2.5° (2) 32.5° 6. bisects TUV. If mTUV 34x and mV... |
all questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 13. Let A and B be two poin... |
of mathematics, Hilbert did succeed in putting geometry on a firm logical foundation. In 1899, Hilbert published a text, Foundations of Geometry, in which he presented a set of axioms that avoided the limitations of Euclid. CHAPTER 3 CHAPTER TABLE OF CONTENTS 3-1 Inductive Reasoning 3-2 Definitions as Biconditionals 3... |
made, results can be only approximate. This is the first weakness in attempting to reach conclusions by inductive reasoning. Use a ruler to draw a pair of parallel line segments by drawing a line segment along opposite edges of the ruler. Turn the ruler and draw another pair of parallel line segments that intersect th... |
from direct measurements of specific cases are only approximate. 3. Care must be taken when applying inductive reasoning to ensure that all relevant examples are examined (no counterexamples exist). 4. Inductive reasoning does not prove or explain conjectures. Exercises Writing About Mathematics 1. After examining sev... |
any parallelogram ABCD, AC BD. 10. In any quadrilateral DEFG, DF bisects D and F. 11. The ray that bisects an angle of a triangle intersects a side of the triangle at its midpoint. 12. Adam made the following statement: “For any counting number n, the expression n2 n 41 will always be equal to some prime number.” He r... |
be 1, 4, and 9. a. Play this same game with cards numbered 1 to 20. What cards are face up when you finish? What property do the numbers on the cards all have in common? b. Play this same game with cards numbered 1 to 30. What cards are face up when you finish? What property do the numbers on the cards all have in com... |
words if and only if, as follows: A triangle is scalene if and only if the triangle has no congruent sides. Every good definition can be written as a true biconditional. Definitions will often be used to prove statements in geometry. EXAMPLE 1 A collinear set of points is a set of points all of which lie on the same s... |
ray is a part of a line that consists of a point on the line, called an endpoint, and all the points on one side of the endpoint. In 9–14, write the biconditional form of each given definition. 9. A point B is between A and C if A, B, and C are distinct collinear points and AB BC AC. 10. Congruent segments are segment... |
� BC, then ABC is a right angle. p → r is true because it is the definition of perpendicular lines. r: ABC is a right angle. r is true by the Law of Detachment. 14365C03.pgs 7/12/07 2:52 PM Page 101 A B A B Deductive Reasoning 101 In the logic-based proof above, notice that the Law of Detachment is cited as a reason fo... |
written in paragraph form, also called a paragraph proof. Each statement must be justified by stating a definition or another 14365C03.pgs 7/12/07 2:52 PM Page 102 102 Proving Statements in Geometry statement that has been accepted or proved to be true. The proof given on page 101 can be written as follows: Proof: We ... |
First use the definition of an angle bisector to prove that the angles are congruent. Then use the definition of congruent angles to prove that the angles have equal measures. Statements is the bisector of ABC. h BD 1. 2. ABD DBC 3. mABD mDBC Exercises Writing About Mathematics Reasons 1. Given. 2. The bisector of an ... |
S Statements Reasons Statements Reasons R T 1. M is the midpoint of. AMB AM 2. MB 3. AM MB 1. 2. 3. 1. RS ST RS > ST 2. 3. RST is isosceles. 1. 2. 3. 15. Given: In ABC, h CE bisects ACB. Prove: mACE mBCE Statements Reasons B E h CE bisects ACB. 1. 2. ACE ECB 3. mACE mECB 1. 2. 3. C A 16. Complete the following proof b... |
proof. Given: ABC is an acute triangle. Prove: mA 90 In this proof, we will use the following definitions: • An acute triangle is a triangle that has three acute angles. • An acute angle is an angle whose degree measure is greater than 0 and less than 90. C A B In the proof, we will use the conditional form of these d... |
ent 1. Assumption. 2. Congruent segments are segments that have the same measure. 3. AB CD 3. Given. 4. CD and AB segments. are not congruent 4. Contradiction in 2 and 3. Therefore, the assumption is false and its negation is true. In this proof, and most indirect proofs, our reasoning reflects the contra- positive of ... |
. If an angle is a right angle, then its degree measure is 90. 4. mCDE 90. 4. Given. 5. CD is not perpendicular to. DE 5. Contradiction in 3 and 4. Therefore, the assumption is false and its negation is true. Exercises Writing About Mathematics 1. If we are given ABC, is it true that intersects g BC? Explain why or why... |
FE and h FG are not opposite rays. a. Write the hypothesis of the conditional as the given. b. Write the conclusion of the conditional as the prove. c. Write an indirect proof for the conditional. 3-5 POSTULATES, THEOREMS, AND PROOF A valid argument that leads to a true conclusion begins with true statements. In Chapt... |
. When we write mP mN, we mean that P and N contain the same number of degrees. Many of our definitions, for example, congruence, midpoint, and bisector, state that two measures are equal. There are three basic properties of equality. The Reflexive Property of Equality: a a The reflexive property of equality is stated ... |
365C03.pgs 7/12/07 2:52 PM Page 112 112 Proving Statements in Geometry Therefore, we can say that “is congruent to” is an equivalence relation on the set of line segments. We will use these postulates of equality in deductive reasoning. In con- structing a valid proof, we follow these steps: 1. A diagram is used to vis... |
intersect to form right angles. 3. A right angle is an angle whose degree measure is 90. 4. Given. 5. Perpendicular lines are two lines that intersect to form right angles. 6. A right angle is an angle whose degree measure is 90. 7. Symmetric property of equality. 8. Transitive property of equality (steps 3 and 7). Ex... |
1. 2. 3. 4. 5. F H J K 14365C03.pgs 7/12/07 2:52 PM Page 115 11. Explain why the following proof is incorrect. B Given: ABC with D a point on Prove: ADB ADC. BC The Substitution Postulate 115 A D C Statements 1. ADB and ADC are right angles. 2. mADB 90 and mADC 90. 3. mADB mADC 4. ADB ADC Reasons 1. Given (from the di... |
3x 2y 13 y x 1 3x 2(x 1) 13 3x 2x 2 13 5x 15 x 3 y 3 1 2 EXAMPLE 1 Given: CE 2CD and CD DE Prove: CE 2DE Proof Statements 1. CE 2CD 2. CD DE 3. CE 2DE Reasons 1. Given. 2. Given. 3. Substitution postulate. (Or: A quantity may be substituted for its equal in any expression of equality.) 14365C03.pgs 7/12/07 2:52 PM Pag... |
When three points, A, B, and C, lie on the same line, the symbol of indicating the following equivalent facts about these points: g ABC is a way • B is on the line segment. AC • B is between A and C. • AB 1 BC 5 AC Since A, B, and C lie on the same line, we can also conclude that into two segments whose sum is. This f... |
stated in symbols or in words as follows: Postulate 3.6 If a b and c d, then a c b d. If equal quantities are added to equal quantities, the sums are equal. The following example proof uses the addition postulate. EXAMPLE 1 Given: g ABC and g DEF with AB DE and BC EF. Prove: AC DF A D B C E F Proof Statements Reasons ... |
mon side between them, and no common interior points. In the diagram, ABC CBD ABD. A B C D 14365C03.pgs 7/12/07 2:52 PM Page 121 The Addition and Subtraction Postulates 121 The Subtraction Postulate The subtraction postulate may also be stated in symbols or in words. Postulate 3.7 If a b, and c d, then a c b d. If equ... |
that the conclusion is valid. B A 3. Given: and AED Prove: AD BC BFC, AE BF, and ED FC Statements Reasons BFC AED 1. and 2. AE ED AD BF FC BC 1. Given. 2. 3. AE BF and ED FC 3. Given. 4. 5. AD BC 4. Substitution postulate. 5. Transitive property (steps 2, 4). D E A C F B 14365C03.pgs 7/12/07 2:52 PM Page 123 The Addit... |
follows: Postulate 3.11 Halves of equal quantities are equal. Note: Doubles and halves of congruent segments and angles will be handled in Exercise 10. Powers Postulate Postulate 3.12 If a b, then a2 b2. The squares of equal quantities are equal. If AB 7, then (AB)2 (7)2, or (AB)2 49. 14365C03.pgs 7/12/07 2:52 PM Page... |
The bisector of an angle is a ray whose endpoint is the vertex of the angle and that divides the angle into two congruent angles. Exercises Writing About Mathematics 1. Explain why the word “positive” is needed in the postulate “Positive square roots of posi- tive equal quantities are equal.” 2. Barry said that “c 0” ... |
. (Think of Main Street as the line segment LPGB.) 10. Explain why the following versions of Postulates 3.9 and 3.11 are valid: Doubles of congruent segments are congruent. Halves of congruent segments are congruent. Doubles of congruent angles are congruent. Halves of congruent angles are congruent. 14365C03.pgs 8/2/0... |
congruent segments are subtracted from congruent segments, the differences are congruent. 3.7.2 If congruent angles are subtracted from congruent angles, the differences are congruent. If equals are multiplied by equals, the products are equal. 3.8 3.9 Doubles of equal quantities are equal. 3.10 If equals are divided ... |
Prove: CM MD CD at M. 8. Given: Prove: is a line segment, RMST RM > ST RM > MS, and MS > ST. 9. Given: Prove ABCD AB > CD is a line segment and AC > BD. 10. Given: SQRP is a line segment and SQ RP. Prove: SR QP 14541C03.pgs 1/25/08 3:51 PM Page 130 130 Proving Statements in Geometry 11. Given: h BC bisects ABD and mCB... |
, then x is equal to (1) 1 (2) 1 (3) 5 (4) 5 2. The property illustrated in the equality 2(a 4) 2(4 a) is (1) the distributive property. (2) the associative property. (3) the identity property. (4) the commutative property. 3. In biconditional form, the definition of the midpoint of a line segment can be written as (1)... |
midpoint of (2) AB BC BD (3) AC CD (4) AC BD AD 9. The statements “AB BC” and “DC BC” are true statements. Which of the following must also be true? (1) AB BC AC (2) A, B, and C are collinear (3) B, C, and D are collinear (4) AB DC 10. Triangle LMN has exactly two congruent sides. Triangle LMN is (3) an isosceles tria... |
the conclusion. a. If x 5, then x 7 5 7. b. If 2y 3 represents a real number, then 2y 3 2y 3. RST is a line segment, then c. If d. If y 2x 1 and y 15, then 2x 1 15. e. If a 3, then RS 1 ST 5 RT. 5 5 3 a 5. 14365C04.pgs 7/12/07 3:04 PM Page 134 CHAPTER 4 CHAPTER TABLE OF CONTENTS 4-1 Postulates of Lines, Line Segments,... |
bush and covered the structure with burlap fabric. During the first winter storm, this protective barrier was pushed out of shape. Her neighbor suggested that she make a tripod of three stakes fastened together at the top, forming three triangles. Melissa found that this arrangement was able to stand up to the storms.... |
P g PD, can be D. P A B Postulate 4.6 From a given point not on a given line, one and only one perpendicular can be drawn to the line. From point P not on drawn perpendicular to perpendicular to g CD. g CD g CD, exactly one line, g PE, can be and no other line from P is P E C D Postulate 4.7 For any two distinct point... |
Page 138 138 Congruence of Line Segments, Angles, and Triangles EXAMPLE 3 h BD bisects ABC and point E is not a point on h BD, can If h BE be the bisector of ABC? Solution No. An angle has one and only bisector. Since point E is not a point on h BE Therefore, h BD h, is not the same ray as BD cannot be the bisector of... |
then be used to help solve numerical and algebraic problems, as in Example 5. EXAMPLE 5 h bisects RST, mRSQ 4x, and SQ mQST 3x 20. Find the measures of RSQ and QST. R Q Solution The bisector of an angle separates the angle into two congruent angles. Therefore, RSQ QST. Then since congruent angles have equal measures, ... |
SR 40 and SQ 15. If mPSQ is twice mQSR and, find mPST. h'SP h'SP, find h SQ Q R S P T mPST. Applying Skills In 16–19, use the given conditional to a. draw a diagram with geometry software or pencil and paper, b. write a given and a prove, c. write a proof. 16. If a triangle is equilateral, then the measures of the side... |
BC CD 1 BC 6. AC > BD 2. Partition postulate. 3. Given. 4. Reflexive property. 5. Addition postulate. 6. Substitution postulate. EXAMPLE 2 Given: M is the midpoint of AB. Prove: AM 1 2AB and MB 1 2AB A M B 14365C04.pgs 7/12/07 3:04 PM Page 142 142 Congruence of Line Segments, Angles, and Triangles M A Proof B Statemen... |
C D 8. If P and T are distinct points and P is, then T is not the RS the midpoint of. midpoint of RS R T P S 9. If DF BE, then DE. BF 10. If AD BC, E is the midpoint of, then BC, AD and F is the midpoint of AE. FC D C E F B A 11. If AC DB AC each other at E, then and and AE DB. EB bisect D A C B E Applying Skills D E ... |
mABC mDEF 4. ABC DEF F B A E D Reasons 1. Given. 2. Definition of right angle. 3. Transitive property of equality. 4. Definition of congruent angles. We can write this proof in paragraph form as follows: Proof: A right angle is an angle whose degree measure is 90. Therefore, mABC is 90 and mDEF is 90. Since mABC and m... |
(90 k) because k (90 k) 90. DEFINITION Supplementary angles are two angles, the sum of whose degree measures is 180. 14365C04.pgs 7/12/07 3:04 PM Page 146 146 Congruence of Line Segments, Angles, and Triangles When two angles are supplementary, each angle is called the supplement of the other. If mc 40 and md 140, the... |
traction postulate. 8. Congruent angles are angles that have the same measure. Note: In a proof, there are two acceptable ways to indicate a definition as a reason. In reason 2 of the proof above, the definition of complementary angles is stated in its complete form. It is also acceptable to indicate this reason by the... |
proofs of Theorems 4.3 and 4.4 and will be left to the student. (See exercises 18 and 19.) More Definitions and Theorems Involving Pairs of Angles DEFINITION A linear pair of angles are two adjacent angles whose sum is a straight angle. C In the figure, ABD is a straight angle and C is not on. Therefore, ABC + CBD ABD... |
ED CEB are a pair of vertical angles because h ED are opposite rays. In each pair of vertical angles, the opposite rays, h EB h EC h EC and and and B which are the sides of the angles, form straight lines, g AB and g CD. When two straight lines intersect, two pairs of vertical angles are formed. 14365C04.pgs 7/12/07 3... |
theorem, in terms of the figure. 3. State the prove, which is the conclusion of the theorem, in terms of the figure. 14365C04.pgs 7/12/07 3:05 PM Page 151 Proving Theorems About Angles 151 4. Present the proof, which is a series of logical arguments used in the demonstration. Each step in the proof should consist of a... |
proved by showing that AEC BED. Do you agree with Josh? Explain. 2. The statement of Theorem 4.7 is “If two angles form a linear pair then they are supplemen- tary.” Is the converse of this theorem true? Justify your answer. Developing Skills In 3–11, in each case write a proof, using the hypothesis as the given and t... |
14365C04.pgs 7/12/07 3:05 PM Page 154 154 Congruence of Line Segments, Angles, and Triangles g RS 16. intersects g LM at P, mRPL x y, mLPS 3x 2y, mMPS 3x – 2y. a. Solve for x and y. b. Find mRPL, mLPS, and mMPS. 17. Prove Theorem 4.2, “If two angles are straight angles, then they are congruent.” 18. Prove Theorem 4.5,... |
ons are named in such a way that each vertex of ABCD corresponds to exactly one vertex of EFGH and each vertex of EFGH corresponds to exactly one vertex of ABCD. This relationship is called a one-to-one correspondence. The order in which the vertices are named shows this one-to-one correspondence of points. 14365C04.pg... |
Page 156 156 Congruence of Line Segments, Angles, and Triangles C F B E A D The correspondence establishes six facts about these triangles: three facts about corresponding sides and three facts about corresponding angles. In the table at the right, these six facts are stated as equalities. Since each congruence statem... |
each other. (Transitive Property) Exercises Writing About Mathematics 1. If ABC DEF, then AB > DE. Is the converse of this statement true? Justify your answer. 2. Jesse said that since RST and STR name the same triangle, it is correct to say RST STR. Do you agree with Jesse? Justify your answer. Developing Skills In 3... |
angle-side measures. (1) 3 in., 90°, 4 in. (2) 5 in., 40°, 5 in. (3) 5 cm, 115°, 8 cm (4) 10 cm, 30°, 8 cm b. For each pair of triangles, measure the side and angles that were not given. Do they have equal measures? c. Are the triangles of each pair congruent? Does it appear that when two sides and the included angle o... |
8. ACD BCD 2. The bisector of a line segment intersects the segment at its midpoint. 3. The midpoint of a line segment divides the segment into two congruent segments. 4. Given. 5. Perpendicular lines intersect to form right angles. 6. If two angles are right angles, then they are congruent. 7. Reflexive property of c... |
SAS. 9. D E A B C 10. C 11. A B A D B E D C 14365C04.pgs 7/12/07 3:05 PM Page 161 Proving Triangles Congruent Using Angle, Side, Angle 161 Applying Skills In 12–14: a. Draw a diagram with geometry software or pencil and paper and write a given state- ment using the information in the first sentence. b. Use the informa... |
and B that are not on g AB. STEP 5. Repeat steps 1 through 4. Let D the vertex of the first angle, E the vertex of the second angle, and F the intersection of the rays of D and E. C A B F D E a. Follow the steps to draw two different triangles with each of the given angle-side-angle measures. (1) 65°, 4 in., 35° (2) 6... |
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