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Name Date6.3Proofs with Parallel LinesFor use with Exploration 6.3Essential Question For which of the theorems involving parallel linesand transversals is the converse true?1EXPLORATION: Exploring ConversesWork with a partner. Write the converse of each conditional statement. Draw adiagram to represent the converse. Determine whether the converse is true. Justifyyour conclusion.a. Corresponding Angles TheoremIf two parallel lines are cut by a transversal, then the pairs ofcorresponding angles are congruent.Converse1 24 35 68 71 24 35 68 7b. Alternate Interior Angles TheoremIf two parallel lines are cut by a transversal, then the pairs ofalternate interior angles are congruent.Conversec. Alternate Exterior Angles TheoremIf two parallel lines are cut by a transversal, then the pairs ofalternate exterior angles are congruent.Converse1931 24 35 68 7Copyright Big Ideas Learning, LLCAll rights reserved.

Name6.31DateProofs with Parallel Lines (continued)EXPLORATION: Exploring Converses (continued)d. Consecutive Interior Angles TheoremIf two parallel lines are cut by a transversal, then the pairs ofconsecutive interior angles are supplementary.ConverseCommunicate Your Answer2. For which of the theorems involving parallel lines and transversals is theconverse true?3. In Exploration 1, explain how you would prove any of the theorems thatyou found to be true.Copyright Big Ideas Learning, LLCAll rights reserved.1941 24 35 68 7

alternate exterior anglesalternate exterior anglesName Dateconsecutive interior e6.3Notetaking with Vocabulary (continued)Date6.3For use after Lesson 6.3TheoremsIn your own words, write the meaning of each vocabulary term.6.3Notetaking with Vocabulary converseCorrespondingAnglesConverseName DateIftwo lines arecut by Anglesa transversalso the alternate interior ConverseIf two lines arecut by a transversalso the corresponding angles2are congruent, then the lines are parallel.45aretwocongruent,thealinesare parallel.6.3withVocabulary(continued)Iflines Notetakingarethencut bytransversalso the alternateinterioranglescorrespondinganglesparallel lines6 2are congruent, then the lines are parallel.45Notes:Name6Alternate Interior Angles Conversej kj kNotes:transversal6.3lines NotetakingIftwoare cut by a transversalso the alternateinterior angleswith Vocabulary(continued)jNotes:j kkarecongruent, then the lines are parallel.45jkjkcongruent then the lines are parallel.congruent,Notes:If two lines are cut by a transversal so the alternate195 exterior angles arecongruent, then the lines are parallel.Notes:195AlternateExterioralternate interiorangles Angles Conversejj k1 kCopyright Big Ideasj Learning, LLCAll rights reserved.j k 1Copyright Big Ideask Learning, LLC8jAll rights reserved.1j kjk8Notes:If two lines are cut by a transversal so the alternate exterior angles areConsecutiveAngles Conversecongruent,then theInteriorlines are parallel.consecutive interior anglesIf two lines are cutInteriorby a transversalso theconsecutive interior angles areConsecutiveAnglesConversesupplementary, then the lines are parallel.Notes:If two lines are cut by a transversal so the consecutive interior angles aresupplementary, then the lines are parallel.1j kjk83j53kjj kNotes:TheoremsConsecutive Interior Angles Converse5Notes:If two lines are cut by a transversal so the consecutive interior angles areTransitive PropertyofareParallelsupplementary,then the linesparallel. Lineskj 8 kIf two lines are cut by a transversal so the alternate exterior angles arecongruent, then the lines are parallel.alternate exteriorangles Angles ConverseAlternateExteriorNotes:j45Notes:Iftwo lines are cut by Anglesa transversalso the consecutive interior angles areCorrespondingConversesupplementary, then the lines are parallel.If two lines are cut by a transversal so the corresponding anglesConsecutiveAnglesare congruent, thenInteriorthe lines areparallel.ConversekkNotes:corresponding anglesIf two lines arecut by a transversalthe alternate exterior angles areAlternateExteriorAngles soConversejkDateAlternate Interior Angles ConverseIf two lines are cut by a transversal so the alternate interior then the linesare parallel.jkIf 3 and 5 aresupplementary,then jj k .32 5 areIf 3 and5jj k.supplementary,then6kk3Notes:k 5pare qIftwo lines arePropertyparallel to theline, thenthey are parallel to each other.If j3 andTransitiveofsameParallelLines5jrsupplementary, then j k .Notes:pq krIf two lines are parallel to the same line, then they are parallel to each other.Notes:Transitive Property of Parallel LinesNotes:If 3 and 5 are k.supplementary,If p q and thenq rj, thenpqrIf two lines are parallel to the same line, then they are parallel to each other.p r.If p q and q r , thenp r. Big Ideas Learning, LLCNotes:CopyrightIf two lines are parallel to the same line, thCopyright Big Ideas Learning, LLC195pAll rights reserved.qr

mA7. j kB7. Corresponding AnglesConverse10. LetA andB be two points on line m. Draw ⃖ ⃗AP and13. yes; Alternate InteriorAngles ConverseNameDateconstruct an angle 1 on n at P so that PAB and 1 arecorrespondingangles. InteriorAnglesConversemP1If twolines are cutAby a transversal so the alternate interior anglesare congruent, thenB the lines are parallel.Example #1Find the value of x that makes m n.Explain your reasoning.m y, an150 n(3x 15) 2Lines m and n are parallel when the marked consecutiveinterior angles are supplementary.180 150 (3x 15) 180 135 3x45 3x45 3x33x 15— —Example #2Find the value of x that makes m n.Explain your reasoning.mn(180 x) kIntegrated Mathematics IWorked-Out Solutionsx 3 and5 arej k.Lines m and n are parallel when the marked alternate exteriorangles are congruent.x (180 x) 2x 1802x 180— —22x 90p Big IdeasLearning,LLC Converse13.Copyrightyes; AlternateInteriorAnglesAll rights reserved.14. yes; Alternate Exterior Angles Converse196.qr 4

Name Date6.3Practice (continued)Extra PracticePracticeAIn Exercises 1 and 2, find the value of x that makes m n. Explain your reasoning.1.2.mn95 m(8x 55) 130 (200 2x) nIn Exercises 3–6, decide whether there is enough information to prove that m n.If so, state the theorem you would use.3.mn4.rrmn5.r6.smmnsnr197Copyright Big Ideas Learning, LLCAll rights reserved.

NameDatePractice10.3 BPractice BIn Exercises 1 and 2, find the value of x that makes s t. Explain your reasoning.r1.s2.t2(x 15) (7x 20) (3x 20) srt(4x 16) In Exercises 3 and 4, decide whether there is enough information to prove thatp q. If so, state the theorem you would use.3.pqr4.x pr(180 x) q5. The map of the United States shows the lines of latitude andlongitude. The lines of latitude run horizontally and the linesof longitude run vertically.aa. Are the lines of latitude parallel? Explain.cbdb. Are the lines of longitude parallel? Explain.7. Given: 1 2 and 2 36. Use the diagram to answer the following.pqrProve: 1 4a(4x 30) (6y) 3(x 1) 6(z 8) b1s432a. Find the values of x, y, and z that makesp q and q r. Explain your reasoning.b. Is p r ? Explain your reasoning.Copyright Big Ideas Learning, LLCAll rights reserved.198cd

yes; m DEB 180 123 57 by the Linear Pair Postulate. So, by defi nition, a pair of corresponding angles are congruent, which means that ⃖AC ⃗ ⃖DF ⃗ by the Corresponding Angles Converse. 22. yes; m BEF 180 37 143 by the Linear Pa

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