3.2 Parallel Lines And Transversals - Static.bigideasmath

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3.2TEXAS ESSENTIALKNOWLEDGE AND SKILLSParallel Lines and TransversalsEssential QuestionWhen two parallel lines are cut by a transversal,which of the resulting pairs of angles are congruent?Exploring Parallel LinesG.5.AG.6.AWork with a partner.Use dynamic geometry softwareto draw two parallel lines. Drawa third line that intersects bothparallel lines. Find the measuresof the eight angles that areformed. What can you conclude?6D54B3E12F68 7524 31USING PRECISEMATHEMATICALLANGUAGEA0 3To be proficient in math,you need to communicateprecisely with others. 2 101234C56Writing ConjecturesWork with a partner. Use the results of Exploration 1 to write conjectures aboutthe following pairs of angles formed by two parallel lines and a transversal.a. corresponding angles1 24 3b. alternate interior angles5 68 71 24 3c. alternate exterior angles1 24 35 68 7d. consecutive interior angles5 68 71 24 35 68 7Communicate Your Answer3. When two parallel lines are cut by a transversal, which of the resulting pairs ofangles are congruent?4. In Exploration 2, m 1 80 . Find the other angle measures.Section 3.2Parallel Lines and Transversals131

What You Will Learn3.2 LessonUse properties of parallel lines.Prove theorems about parallel lines.Core VocabulVocabularylarrySolve real-life problems.Previouscorresponding anglesparallel linessupplementary anglesvertical anglesUsing Properties of Parallel LinesTheoremsTheorem 3.1 Corresponding Angles TheoremIf two parallel lines are cut by a transversal, then the pairs of correspondingangles are congruent.t1 23 4Examples In the diagram at the left, 2 6 and 3 7.ProofEx. 36, p. 184pTheorem 3.2 Alternate Interior Angles Theorem5 67 8If two parallel lines are cut by a transversal, then the pairs of alternate interiorangles are congruent.qExamples In the diagram at the left, 3 6 and 4 5.ProofExample 4, p. 134Theorem 3.3 Alternate Exterior Angles TheoremIf two parallel lines are cut by a transversal, then the pairs of alternate exteriorangles are congruent.Examples In the diagram at the left, 1 8 and 2 7.ProofEx. 15, p. 136Theorem 3.4 Consecutive Interior Angles TheoremIf two parallel lines are cut by a transversal, then the pairs of consecutive interiorangles are supplementary.Examples In the diagram at the left, 3 and 5 are supplementary, and 4 and 6 are supplementary.ANOTHER WAYThere are many waysto solve Example 1.Another way is to use theCorresponding AnglesTheorem to find m 5and then use the VerticalAngles CongruenceTheorem (Theorem 2.6)to find m 4 and m 8.ProofEx. 16, p. 136Identifying AnglesThe measures of three of the numbered angles are120 . Identify the angles. Explain your reasoning.SOLUTION120º 23 45 67 8By the Alternate Exterior Angles Theorem, m 8 120 . 5 and 8 are vertical angles. Using the Vertical Angles Congruence Theorem(Theorem 2.6), m 5 120 . 5 and 4 are alternate interior angles. By the Alternate Interior Angles Theorem, 4 120 .So, the three angles that each have a measure of 120 are 4, 5, and 8.132Chapter 3Parallel and Perpendicular Lines

Using Properties of Parallel LinesFind the value of x.115 4(x 5) abSOLUTIONBy the Vertical Angles Congruence Theorem (Theorem 2.6), m 4 115 . Lines a andb are parallel, so you can use the theorems about parallel lines.Check115 (x 5) 180 ?115 (60 5) 180180 180m 4 (x 5) 180 Consecutive Interior Angles Theorem115 (x 5) 180 Substitute 115 for m 4.x 120 180 Combine like terms.x 60Subtract 120 from each side.So, the value of x is 60.Using Properties of Parallel LinesFind the value of x.1136 c(7x 9) dSOLUTIONBy the Linear Pair Postulate (Postulate 2.8), m 1 180 136 44 . Lines c and dare parallel, so you can use the theorems about parallel lines.m 1 (7x 9) Check44 (7x 9) ?44 7(5) 944 44 Alternate Exterior Angles Theorem44 (7x 9) Substitute 44 for m 1.35 7xSubtract 9 from each side.5 xDivide each side by 7.So, the value of x is 5.Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comUse the diagram.1. Given m 1 105 , find m 4, m 5, andm 8. Tell which theorem you use in each case.2. Given m 3 68 and m 8 (2x 4) ,1 23 45 67 8what is the value of x? Show your steps.Section 3.2Parallel Lines and Transversals133

Proving Theorems about Parallel LinesProving the Alternate Interior Angles TheoremProve that if two parallel lines are cut by a transversal, then the pairs of alternateinterior angles are congruent.SOLUTIONSTUDY TIPBefore you write a proof,identify the Given andProve statements for thesituation described or forany diagram you draw.tDraw a diagram. Label a pair of alternateinterior angles as 1 and 2. You are looking foran angle that is related to both 1 and 2. Noticethat one angle is a vertical angle with 2 and acorresponding angle with 1. Label it 3.p12q3Given p qProve 1 2STATEMENTSREASONS1. p q1. Given2. 1 32. Corresponding Angles Theorem3. 3 23. Vertical Angles Congruence Theorem (Theorem 2.6)4. 1 24. Transitive Property of Congruence (Theorem 2.2)Monitoring ProgressHelp in English and Spanish at BigIdeasMath.com3. In the proof in Example 4, if you use the third statement before the secondstatement, could you still prove the theorem? Explain.Solving Real-Life ProblemsSolving a Real-life ProblemWhen sunlight enters a drop of rain,different colors of light leave the dropat different angles. This process iswhat makes a rainbow. For violet light,m 2 40 . What is m 1? How doyou know?21SOLUTIONBecause the Sun’s rays are parallel, 1 and 2 are alternate interior angles.By the Alternate Interior Angles Theorem, 1 2.So, by the definition of congruent angles, m 1 m 2 40 .Monitoring ProgressHelp in English and Spanish at BigIdeasMath.com4. WHAT IF? In Example 5, yellow light leaves a drop at an angle of m 2 41 .What is m 1? How do you know?134Chapter 3Parallel and Perpendicular Lines

Exercises3.2Dynamic Solutions available at BigIdeasMath.comVocabulary and Core Concept Check1. WRITING How are the Alternate Interior Angles Theorem (Theorem 3.2) and the Alternate ExteriorAngles Theorem (Theorem 3.3) alike? How are they different?2. WHICH ONE DOESN’T BELONG? Which pair of angle measures does not belong with theother three? Explain.m 1 and m 3m 2 and m 4m 2 and m 3m 1 and m 512435Monitoring Progress and Modeling with Mathematics10.In Exercises 3–6, find m 1 and m 2. Tell whichtheorem you use in each case. (See Example 1.)3.4.117 (8x 6) 150 11225.6.1 2140 122 118 421In Exercises 11 and 12, find m 1, m 2, and m 3.Explain your reasoning.11.1 280 3In Exercises 7–10, find the value of x. Show your steps.(See Examples 2 and 3.)7.12.18.2x 128 72 (7x 24) 133 2313. ERROR ANALYSIS Describe and correct the error inthe student’s reasoning.9.65º5 (11x 17)º Section 3.2910 9 10 bythe CorrespondingAngles Theorem(Theorem 3.1).Parallel Lines and Transversals135

A14. HOW DO YOU SEE IT?Da.b.19. CRITICAL THINKING Is it possible for consecutiveBUse the diagram.interior angles to be congruent? Explain.C— andName two pairs of congruent angles when AD—BC are parallel. Explain your reasoning.—Name two pairs of supplementary angles when AB—and DC are parallel. Explain your reasoning.PROVING A THEOREM In Exercises 15 and 16, prove thetheorem. (See Example 4.)15. Alternate Exterior Angles Theorem (Thm. 3.3)20. THOUGHT PROVOKING The postulates and theoremsin this book represent Euclidean geometry. Inspherical geometry, all points are points on the surfaceof a sphere. A line is a circle on the sphere whosediameter is equal to the diameter of the sphere. Inspherical geometry, is it possible that a transversalintersects two parallel lines? Explain your reasoning.MATHEMATICAL CONNECTIONS In Exercises 21 and 22,write and solve a system of linear equations to find thevalues of x and y.21.16. Consecutive Interior Angles Theorem (Thm. 3.4)17. PROBLEM SOLVINGA group of camperstie up their foodbetween twoparallel trees, asshown. The rope ispulled taut, forminga straight line.Find m 2. Explainyour reasoning.(See Example 5.)(14x 10) 22.2y 4x 2y (2x 12) (y 6) 5x 23. MAKING AN ARGUMENT During a game of pool,your friend claims to be able to make the shotshown in the diagram by hitting the cue ball sothat m 1 25 . Is your friend correct? Explainyour reasoning.76 2118. DRAWING CONCLUSIONS You are designing a boxlike the one shown.65 123A1B3 2C— bisects24. REASONING In the diagram, 4 5 and SE RSF. Find m 1. Explain your reasoning.a. The measure of 1 is 70 . Find m 2 and m 3.EFb. Explain why ABC is a straight angle.c. If m 1 is 60 , will ABC still be a straight angle?Will the opening of the box be more steep or lesssteep? Explain.Maintaining Mathematical Proficiency41T23Reviewing what you learned in previous grades and lessonsWrite the converse of the conditional statement. Decide whether it is true or false. (Section 2.1)25. If two angles are vertical angles, then they are congruent.26. If you go to the zoo, then you will see a tiger.27. If two angles form a linear pair, then they are supplementary.28. If it is warm outside, then we will go to the park.136Chapter 3Parallel and Perpendicular Lines5SR

Section 3.2 Parallel Lines and Transversals 133 Using Properties of Parallel Lines Find the value of x. a b 4 115 (x 5) SOLUTION By the Vertical Angles Congruence Theorem (Theorem 2.6), m 4 115 .Lines a and b are parallel, so you can use the theorems about parallel lines. m 4 (x 5) 180 Consecutive Interior An

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