2.6 Proving Statements About Angles
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Page 2 of 8EXAMPLE 2ProofUsing the Transitive PropertyThis two-column proof uses the Transitive Property.1GIVEN m 3 40 , 1 2, 2 32PROVE m 1 40 m 3 40 , 1 2, 2 3 1 3m 1 m 3m 1 40 43GivenTransitive Property of CongruenceDefinition of congruent anglesSubstitution property of equalityTHEOREMTHEOREM 2.3All right angles are congruent.EXAMPLE 3ProofProving Theorem 2.3You can prove Theorem 2.3 as shown.GIVEN 1 and 2 are right angles12PROVE 1 2 1 and 2 are right anglesm 1 90 , m 2 90 m 1 m 2 1 2GivenDefinition of right angleTransitive property of equalityDefinition of congruent anglesACTIVITYUsingTechnologyInvestigating Supplementary AnglesUse geometry software to draw and label two intersecting lines.1101What do you notice about the measuresof AQB and AQC? AQC and CQD? AQB and CQD?2Rotate BC to a different position. Do theangles retain the same relationship?3Make a conjecture about two anglessupplementary to the same angle. Chapter 2 Reasoning and Proofq
Page 3 of 8GOAL 2PROPERTIES OF SPECIAL PAIRS OF ANGLESTHEOREMSTHEOREM 2.4If two angles are supplementary tothe same angle (or to congruent angles)then they are congruent.If213 1 2 180 and 2 3 180 , then 1 3.THEOREM 2.5If two angles are complementary to thesame angle (or to congruent angles) thenthe two angles are congruent.IfEXAMPLE 4Proof4 4 5 90 and 5 6 90 , then 4 6.5634Proving Theorem 2.4GIVEN 1 and 2 are supplements, 3 and 4 are supplements, 1 412PROVE 2 3 1 and 2 are supplements, 3 and 4 are supplements, 1 4m 1 m 2 180 m 3 m 4 180 m 1 m 2 m 3 m 4m 1 m 4m 1 m 2 m 3 m 1m 2 m 3 2 3GivenDefinition of supplementaryanglesTransitive property of equalityDefinition of congruent anglesSubstitution property of equalitySubtraction property of equalityDefinition of congruent anglesP O S T U L AT EPOSTULATE 12If two angles form a linear pair,then they are supplementary.12m 1 m 2 180 2.6 Proving Statements about Angles111
Page 4 of 8EXAMPLE 5Using Linear PairsIn the diagram, m 8 m 5 and m 5 125 .Explain how to show m 7 55 .5 67 8SOLUTIONUsing the transitive property of equality, m 8 125 . The diagram showsm 7 m 8 180 . Substitute 125 for m 8 to show m 7 55 .THEOREMTHEOREM 2.6241Vertical angles are congruent.3 1 3, 2 4EXAMPLE 6STUDENT HELPStudy TipRemember that previouslyproven theorems can beused as reasons in aproof, as in Step 3 of theproof at the right.Proving Theorem 2.6GIVEN 5 and 6 are a linear pair, 6 and 7 are a linear pair5PROVE 5 7 5 and 6 are a linear pair, 6 and 7 are a linear pair 5 and 6 are supplementary, 6 and 7 are supplementary 5 767GivenLinear Pair PostulateCongruent Supplements TheoremGUIDED PRACTICEVocabulary CheckConcept Check ? and QRS XYZ, then CDE XYZ,” is an1. “If CDE ? Property of Angle Congruence.example of the 2. To close the blades of the scissors, you close thehandles. Will the angle formed by the blades be thesame as the angle formed by the handles? Explain.Skill Check 3. By the Transitive Property of Congruence,? C.if A B and B C, then In Exercises 4–9, 1 and 3 are a linear pair, 1 and 4 are a linear pair,and 1 and 2 are vertical angles. Is the statement true?1124. 1 35. 1 26. 1 47. 3 28. 3 49. m 2 m 3 180 Chapter 2 Reasoning and Proof
Page 5 of 8PRACTICE AND APPLICATIONSSTUDENT HELP10.Extra Practiceto help you masterskills is on p. 806.PROVING THEOREM 2.2 Copy and complete the proof of theSymmetric Property of Congruence for angles.GIVEN A BPROVE B A? Definition of congruent angles? ? A B? m B m A B A11.PROVING THEOREM 2.2 Write a two-column proof for the ReflexiveProperty of Congruence for angles.FINDING ANGLES In Exercises 12–17, complete the statement given thatm EHC m DHB m AHB 90 ? .12. If m 7 28 , then m 3 ? .13. If m EHB 121 , then m 7 71 6? .14. If m 3 34 , then m 5 54? .15. If m GHB 158 , then m FHC 3? .16. If m 7 31 , then m 6 ? .17. If m GHD 119 , then m 4 18.PROVING THEOREM 2.5 Copy and complete the proof of theCongruent Complements Theorem.GIVEN 1 and 2 are complements, 3 and 4 are complements, 2 42PROVE 1 31. 1 and 2 are complements,STUDENT HELPHOMEWORK HELPExample 1:Example 2:Example 3:Example 4:Example 5:Example 6:Exs. 10, 11Exs. 12–17Exs. 12–17Exs. 19–22Exs. 23–28Exs. 23–2831 3 and 4 are complements, 2 4? , ? 2. 3. m 1 m 2 m 3 m 44. m 2 m 45. m 1 m 2 m 3 m 26. m 1 m 3? 7. 4? 1. 2. Def. of complementary angles3. Transitive property of equality? 4. ? 5. ? 6. 7. Definition of congruent angles2.6 Proving Statements about Angles113
Page 6 of 8FINDING CONGRUENT ANGLES Make a sketch using the given information.Then, state all of the pairs of congruent angles.19. 1 and 2 are a linear pair. 2 and 3 are a linear pair. 3 and 4 are alinear pair.20. XYZ and VYW are vertical angles. XYZ and ZYW are supplementary. VYW and XYV are supplementary.21. 1 and 3 are complementary. 4 and 2 are complementary. 1 and 2are vertical angles.22. ABC and CBD are adjacent, complementary angles. CBD and DBFare adjacent, complementary angles.WRITING PROOFS Write a two-column proof.INTSTUDENT HELPNEER THOMEWORK HELPVisit our Web sitewww.mcdougallittell.comfor help with Exs. 23–26.23. GIVEN m 3 120 , 1 4,24. GIVEN 3 and 2 are 3 4complementary,m 1 m 2 90 PROVE m 1 120 PROVE 3 1Plan for Proof First show that 1 3. Then use transitivity toshow that m 1 120 .Plan for Proof First show that 1 and 2 are complementary.Then show that 3 1.32415261326. GIVEN 5 625. GIVEN QVW and RWV aresupplementaryPROVE 4 7PROVE QVP RWVPlan for Proof First show thatPlan for Proof First show that QVPand QVW are supplementary. Thenshow that QVP RWV. 4 5 and 6 7. Thenuse transitivity to show that 4 7.q456xy USING ALGEBRA In Exercises 27 and 28, solve for each variable.Explain your reasoning.27.28.(4w 10) 13w 2(x 25) (2x 30) 3y (10z 45) 3(6z 7) (4y 35) 114Chapter 2 Reasoning and Proof7
Page 7 of 8FOCUS ONAPPLICATIONSREFELAL I29.WALL TRIM A chair rail is a type of wall trim that is placed about threefeet above the floor to protect the walls. Part of the chair rail below has beenreplaced because it was damaged. The edges of the replacement piece wereangled for a better fit. In the diagram, 1 and 2 are supplementary, 3and 4 are supplementary, and 2 and 3 each have measures of 50 . Is 1 4? Explain.1 2MITER BOX This3 4box has slotted sidesto guide a saw when makingangled cuts.PICTURE FRAMES Suppose you are making30.4a picture frame, as shown at the right. The cornersare all right angles, and m 1 m 2 52 .Is 4 3? Explain why or why not.31.123Writing Describe some instances of mitered,or angled, corners in the real world.TECHNOLOGY Use geometry software to draw two overlapping right32.angles with a common vertex. Observe the measures of the three anglesas one right angle is rotated about the other. What theorem does this illustrate?TestPreparationQUANTITATIVE COMPARISON Choose the statement that is true about thediagram. In the diagram, 9 is a right angle and m 3 42 .A¡B¡C¡D¡The quantity in column A is greater.The quantity in column B is greater.The two quantities are equal.The relationship can’t be determinedfrom the given information.Column A Challenge71 283456Column B33.m 3 m 4m 1 m 234.m 3 m 6m 7 m 835.m 53(m 3)36.m 7 m 8m 937.9PROOF Write a two-column proof.GIVEN m ZYQ 45 ,m ZQP 45 PROVE ZQR XYQq2.6 Proving Statements about Angles115
Page 8 of 8MIXED REVIEWFINDING ANGLE MEASURES In Exercises 38–40, the measure of 1 and therelationship of 1 to 2 is given. Find m 2. (Review 1.6 for 3.1)38. m 1 62 , complementary to 239. m 1 8 , supplementary to 240. m 1 47 , complementary to 241. PERPENDICULAR LINES The definition of perpendicular lines states that iftwo lines are perpendicular, then they intersect to form a right angle. Is theconverse true? Explain. (Review 2.2 for 3.1)xy USING ALGEBRA Use the diagram and the given information to solvefor the variable. (Review 2.5)ÆÆ ÆÆÆÆ ÆÆÆÆ ÆÆÆÆ Æ16x 542. AD EF, EF CF43. AB EF, EF BC28x 11w 21.5y3w 43z 29z44. DE EF, EF JKÆ5y 745. JM ML, ML KLQUIZ 2Solve the equation and state a reason for each step. (Lesson 2.4)1. x º 3 72. x 8 273. 2x º 5 134. 2x 20 4x º 125. 3(3x º 7) 66. º2(º2x 4) 16PROOF In Exercises 7 and 8 write a two column proof. (Lesson 2.5)ÆÆ ÆÆÆÆÆÆ7. GIVEN BA BC, BC CD,AE DFPROVE BE CF9.116Æ ÆÆÆÆPROVE FG EHASTRONOMY While lookingthrough a telescope one night, you beginlooking due east. You rotate the telescopestraight upward until you spot a comet.The telescope forms a 142 angle withdue east, as shown. What is the angle ofinclination of the telescope from duewest? (Lesson 2.6)Chapter 2 Reasoning and ProofÆ8. GIVEN EH GH, FG GHWest142 East
2.6 Proving Statements about Angles 111 PROPERTIES OF SPECIAL PAIRS OF ANGLES Proving Theorem 2.4 GIVEN 1 and 2 are supplements, 3 and 4 are supplements, 1 4 PROVE 2 3 EXAMPLE 4 GOAL 2 POSTULATE 12 Linear Pair Postulate If two angles form a
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2.7 Statements 2.7.1 Expression Statements and Compound Statements Expression Statement: Similar to C, classical statements are expression statements. Compound Statement: A group of statements could be chained back to back, but only when expected by conditional statements if, while or for. Collectively we can refer those as statement lists.
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As calls are made for reasoning-and-proving to permeate school mathematics, several textbook analyses have been conducted to identify reasoning-and-proving opportunities outside of high-school geometry. This study loo
Aug 27, 2015 · 4.3 Proving Triangles are Congruent: SSS and SAS 213 The SSS Congruence Postulate is a shortcut for proving two triangles are congruent without using all six pairs of corresponding parts. The postulate below is a shortcut that uses two sides and the angle that is includedbetween the sides. Using the SAS Congruence Postulate Prove that AEB .
Section 9.4 Proving Statements about Segments and Angles 469 Modeling with Mathematics A park, a shoe store, a pizza shop, and a movie theater are located in order on a city street. The distance between the park and the shoe store is the same as the distance between the pizza shop
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