3.2 Lesson What You Will Learn English Language Learners Use properties of parallel lines. Prove theorems about parallel lines.
Notebook Development A Is it possible for consecutive Core VocabularyVocabulary Solve real-life problems.14. HOW DO YOU SEE IT? B 19. CRITICAL THINKING Use the diagram. interior angles to be congruent? Explain. Have students record Theorems 3.1, PreviousDynamicDynamic TeTeachingaching ToolsTools D DynamicDynamic Assessment3.2 & ProProgressgress LessonMonitorinMonitoringg TooTooll What You Will Learn C The postulates and theorems English Language Learners corresponding angles 20. THOUGHT PROVOKING 3.2, 3.3, and 3.4 in their notebooks. Use properties of parallel lines. in this book represent Euclidean geometry. In InteractiveInteractive Whiteboard Lesson LibrarLibraryyUsing Properties a. of Name Parallel two pairs of congruent Lines angles when AD— and Prove theorems about parallel lines. spherical geometry, all points are points on the surface Notebook Development parallel lines BC — are parallel. Explain your reasoning. Include a sketch, the full theorem DynamicDynamic CClassroomlassroom witwithCoreh DDynamicy namicVocabularyVocabu InvestiInvestigationslagrationy s Solve real-life problems. of a sphere. A line is a circle on the sphere whose Have students record Theorems 3.1, supplementary anglesPrevious b. Name two pairs of supplementary angles when AB— diameter is equal to the diameter of the sphere. In 3.2, 3.3, and 3.4 in their notebooks. corresponding angles name, and an example for each Usingand Properties DC — are parallel. Explain of Parallel your reasoning. Lines spherical geometry, is it possible that a transversal Include a sketch, the full theorem verticalPROOFS angles WITHparallel linesPARALLELTheorems LINES (3.3) supplementary angles intersects two parallel lines? Explain your reasoning. name, and an example for each ANSWERS theorem. vertical angles Theorems theorem. 14. a. ADB ≅ CBD and Theorem 3.1 CorrespondingTheorem 3.1 Corresponding Angles Theorem Angles Theorem PROVING A THEOREM In Exercises 15 and 16, prove the MATHEMATICAL CONNECTIONS In Exercises 21 and 22, CAD ≅ ACB by the Alternate theorem.If two parallel(See Example lines 4.)are cut by a transversal, then the pairs of corresponding If two parallel linesangles are cut are by congruent. a transversal, then the pairs of correspondingwrite and solve a system of linear equations to f nd the Interior Angles Theorem t values of x and y. Alternate Exterior Angles Theorem (Thm. 3.3) Extra Example 1 (Thm. 3.2). angles are congruent.15.Examples In the diagram at the left, 2 ≅ 6 and 3 ≅ 7. The measures of three of the numbered 12 Proof Ex. 36, p. 180 21. (14x − 10)° 22. 2y° 4x° b. BAD andt CDA are 43 p 16. Consecutive Interior Angles Theorem (Thm. 3.4) angles are 75°. Identify the angles. Explain Examples In the diagramTheorem at 3.2the left,Alternate 2 ≅ 6 Interior and 3 ≅Angles 7. Theorem Extra Example 1 supplementary, as well as (2x 12) (y 6) your reasoning. 17.If twoPROBLEM parallel SOLVING lines are cut by a transversal, then the pairs of 2alternatey° 5x° interior + ° + ° ABC and DCB, by the angles are congruent. 12 56Proof Ex. 36, p. 180 A group of campers The measures of three of the numbered t Consecutive Interior Angles p 87 q Examplestie up their foodIn the diagram at the left, 3 ≅ 6 and 4 ≅ 5. 12 Theorem43 (Thm. 3.4). Proofbetween Example two 4, p. 134 23. MAKING AN ARGUMENT During a game of pool, parallel trees, as angles are 75°. Identify the angles. Explain75° 4 p 15–16. See Additional Answers. your friend claims to be able to make the shot Theorem 3.2 AlternateTheoremshown. The 3.3Interior rope isAlternate Angles Exterior Theorem Angles Theoremshown in the diagram by hitting the cue ball so 76 If twopulled parallel taut, forming lines are cut by a transversal,° then the pairsthat of alternatem 1 25 exterior. Is your friend correct? Explain 17. m 2 = 104°; Because the trees 2 = ° your reasoning. anglesa straight are congruent.line. your reasoning. 5 6 form parallel lines, and the ropeIf is two a parallel lines are cut by a transversal, then the pairs of alternate interior ExamplesFind m 2. ExplainIn the diagram at the left, 1 ≅ 8 and 2 ≅ 7. q 78 transversal, the 76° angle and angles2 are are congruent.Proofyour reasoning.Ex. 15, p. 136 t consecutive56 interior angles. So, they (See Example 5.) are supplementary87 q by the ConsecutiveExamples In the diagramTheorem at 3.4the left,Consecutive 3 ≅ 6 and Interior 4 ≅ Angles5. Theorem 1 2, 6, 7; Sample answer: By the 18.If twoDRAWING parallel CONCLUSIONS lines are cut You by are a transversal,designing a box then the pairs of consecutive interior Interior Angles Theorem (Thm. 3.4). Vertical Angles Congruence Theorem Proof Example 4, anglesp. 134like theare one supplementary. shown. 12(Thm. 2.6), m 2 = 75°. By the Alternate 18–28. See Additional Answers. Examples In the diagram at the left, 3 and 5 are supplementary, and 4 and 6 are supplementary. Interior Angles Theorem (Thm. 3.2), ANOTHER WAY 65° 75° 4 p There are many ways Proof Ex. 16, p. 136 m 6 = 75°. By the Corresponding Angles to solve Example 1. Theorem (Thm. 3.1), m 7 = 75°. ______Another way is toTheorem use the 3.3 Alternate Exterior AnglesA Theorem Corresponding Angles 1 2 1 B Identifying Angles — Theorem to f nd m 5 3 3 2 C 24. REASONING In the diagram, 4 ≅ 5 and SE bisects and then use theIf Vertical two parallel lines are cut by a transversal, then the pairs of alternate exteriorRSF . Find m 1. Explain your reasoning. The measures of three of the numbered angles are Angles Congruence 120°. Identify the angles. Explain your reasoning. WARM UP: Theorem (Theorem angles 2.6) are congruent. E Mini-Assessment a. The measure of 1 is 70 . Find m 2 and m 3. 120º 52 6 to f nd m 4 and m 8. ° 43 87 5 6 SOLUTION F 4 Examples In the diagram b. Explain at the why left, ABC 1is ≅a straight 8 and angle. 2 ≅ 7. Use the diagram. By the Alternate Exterior Angles Theorem, m 8 = 120°. DISCUSSION c. If m 1 is 60°, will ABC still be a straight angle? 2 78 q 5 and 8 are vertical angles. Using the Vertical Angles Congruence Theorem 1 3 5 more steep less Proof Ex. 15, p.(Theorem 136 Will 2.6), the mopening 5 = of 120 the° .box be or TRS steep? Explain. 5 and 4 are alternate interior angles. By the Alternate Interior Angles Theorem, 12 56 4 = 120°. Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons 34 78 Theorem 3.4 Consecutive So, the three anglesInterior that eachAngles have Theorema measure of 120° are 4, 5, and 8. 132 Chapter 3 Parallel and PerpendicularWrite the converse Lines of the conditional statement. Decide whether it is true or false. (Section 2.1) 2, 6, 7; Sample answer: By the If two parallel lines are 25. cut If twoby aangles transversal, are vertical then angles, the then pairs they of are consecutive congruent. interior
1. Given m 7 = 72°, find m 2 angles are supplementary. 26. If you go to the zoo, then you will see a tiger. Vertical Angles Congruence Theorem ! hs_geo_pe_0302.indd 132 ! ! 1/19/15 9:23 AM and m 5. mm ∠2 =7 72= °7; 2 m∠2 27.m ∠If two5 angles form a linearm pair,∠ then8 = they11 are5 supplementary.m∠2 = (2x − 3) 1. If theLaurie’s Notes, findExamples Teacher In theand diagram Actions at .the left, 2. If3 and 5 are supplementary,and and , find the value of x. (Thm. 2.6), m 2 75 . By the Alternate m 5 = 108° 28. If it is warm outside, then we will go to the park. = ° • Write the theorems. 4 and 6 are supplementary. ANOTHER2. Given WAY m 8 = 115° and Interior Angles Theorem (Thm. 3.2), Probing Question: “Will there136 always Chapter be four 3 acute Parallel and and fourPerpendicular obtuse Lines angles when the There arem many2 = (2 transversalwaysx − 3)°. Find intersects the valueProof the Ex. two 16, parallel p. 136 lines? Explain.” no; There could be eight right angles if m 6 = 75°. By the Corresponding Angles of x. 34 the transversal is perpendicular to the parallel lines. to solve Example• MP3 1. Construct Viable Arguments and Critique the Reasoning of Others: Students 3. A bicycleneed path divides to justify a rectangular why each angle has a measure of 120° in Example 1. There are different trains Another way is to use the hs_geo_pe_0302.indd 136 1/19/15 9:23 AM Theorem (Thm. 3.1), m 7 = 75°. park. Theof path logic makes that a 42students° angle may follow. Give time for partners to discuss before having the whole-class Corresponding Anglesdiscussion. If students need help... If students got it... with the top border of the park, so Identifying Angles Theorem to f nd m 5 m 1 = 42°. What is m 2? How do Resources by Chapter Resources by Chapter and thenyou use know? the Vertical ______The measures of three• Practice of the numberedA and Practice angles B are • Enrichment and Extension Angles Congruence • Puzzle Time • Cumulative Review 132 Chapter 3 PROOFS: Corresponding1 120°. Identify Angles the angles. Postulate Explain your reasoning. Theorem (Theorem 2.6) Same Side InteriorStudent Angle Journal Theorem (Consecutive Interior120º Angle52 6 Theorem) Start the next Section to f nd m 4 and m 8. Same Side Exterior• Angle Practice Theorem 43 87 SOLUTION hscc_geo_te_0302.indd 132 Alternate Interior DifferentiatingAngle Theorem the Lesson 2/12/15 2:04 PM 2 AlternateBy the Exterior Alternate Skills ExteriorAngle Review Angles Theorem Handbook Theorem, m 8 = 120°. m 2 = 138°; Consec. Int. ⦞ Theorem (Thm. 3.4) 5 and 8 are vertical angles. Using the Vertical Angles Congruence Theorem (Theorem 2.6), m 5 = 120°.
136 Chapter 3 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. hscc_geo_te_0302.indd 136 2/12/15 2:04 PM 132 Chapter 3 Parallel and Perpendicular Lines
hs_geo_pe_0302.indd 132 1/19/15 9:23 AM Laurie’s Notes Teacher Actions • Write the theorems. Probing Question: “Will there always be four acute and four obtuse angles when the transversal intersects the two parallel lines? Explain.” no; There could be eight right angles if the transversal is perpendicular to the parallel lines. • MP3 Construct Viable Arguments and Critique the Reasoning of Others: Students need to justify why each angle has a measure of 120° in Example 1. There are different trains of logic that students may follow. Give time for partners to discuss before having the whole-class discussion.
132 Chapter 3
hscc_geo_te_0302.indd 132 2/12/15 2:04 PM Extra Example 1 3.3 Lesson What You Will Learn Find the value of x that makes m ʈ n. Use the Corresponding Angles Converse. Construct parallel lines. Core VocabularyVocabulary Prove theorems about parallel lines. 72 ° Previous Use the Transitive Property of Parallel Lines. converse m parallel lines transversal Using the Corresponding Angles Converse 4(x + 5)° n corresponding angles Theorem 3.5 below is the converse of the Corresponding Angles Theorem (Theorem congruent 3.1). Similarly, the other theorems about angles formed when parallel lines are cut by alternate interior angles a transversal have true converses. Remember that the converse of a true conditional statement is not necessarily true, so you must prove each converse of a theorem. x = 22 alternate exterior angles consecutive interior angles Theorem MONITORINGConstructingWhat You Will PROGRESS ParallelLearn Lines Extra Example 1 3.3 Lesson Theorem 3.5 Corresponding Angles Converse ANSWERSThe Corresponding Angles Converse justif es the construction of parallel lines, Find the value of x that makes m ʈ n. What Use You the Will Corresponding Learn Angles Converse. If two lines are cut Englishby a transversal Language so the Learners Extra Example 1 3.3 Lesson 1.as shownyes; The below. angle that is corresponding corresponding angles are congruent, then 2 Construct parallel lines. Find the value of x that makes m ʈ n. Usewith the the Corresponding 75° angle also Angles forms Converse. a the lines are parallel. j Core VocabularyVocabulary Prove theorems about parallel lines. 6 Constructlinear pair parallel with thelines. 105° angle. So, Pair Activity Previous k 72 ° Core VocabularyVocabulary Proveit must Use theorems the be Transitive180 °about − 105 Property parallel° = Constructing 75 lines.of° byParallel Lines. Parallel Lines Proof Ex. 36, p. 180Have students work in pairs to perform converse j ʈ k 72 ° m Previous Usethe theLinear Transitive Pair Postulate Property of (Post. Parallel 2.8). Lines. converseparallel lines Use a compass and straightedge to construct a line the construction Constructing Parallel m UsingBecause the the Correspondingcorresponding angles Angles Converse P paralleltransversal lines through point P that is parallel to line m. Constructing UsingLines. the As Corresponding Parallelthey construct Lines Angles a line Converse through transversalcorresponding angles UsingTheoremhave the the 3.5 same Correspondingbelow measure,is the converse they of areAngles the Corresponding Converse Angles Theorem (Theorem 4(x + 5)° n m The Corresponding Angles Converse justif es the construction of parallel lines, English Language Learners 4(x 5) n correspondingcongruent angles Theorem3.1).congruent Similarly, 3.5 below by the is dethe otherf converse nition. theorems ofSo, the about m Corresponding ʈ nangles by formed Angles when Theorem parallel (Theorem lines are cut by Find the value ofas xshown thata makes below.point m ʈ thatn . is parallel to a given line, + ° (3x + 5)° congruent 3.1).a transversalSimilarly, the have other true theorems converses. about Remember angles formed that the when converse parallel of lines a true are conditional cut by alternate interior angles SOLUTIONthe Corresponding Angles Converse students should take turns readingm alternate interior angles aConstructing transversalstatement haveis not true necessarily Parallel converses. true, Remember Lines so you mustthat the prove converse each converseof a true conditionalof a theorem. Pair Activity alternate exterior angles statement(Thm. is not 3.5). necessarily true, so you must prove each converse of a theorem. 65° x = 22 alternate exterior angles The Corresponding Angles Converse justif es the construction of parallel lines, the instructions Constructing aloud and Parallel executing Lines Have students work in pairs to perform x = 22 Step 1consecutive interior angles Step2. 2CONVERSE The hypothesis OF and conclusionPARALLEL Step 3 of LINES THEOREMS Step 4 English Language Learners n consecutive interior angles as shown below. the steps. the construction Constructing Parallel TheoremtheTheorem Corresponding Angles Converse SOLUTION Use a compass and straightedge to construct a line P C C Pair Activitythrough point P that is parallel to line m. Lines. As they construct a line through MONITORINGMONITORING PROGRESS (Thm.C 3.5) are the reverse of the Lines m and n are parallel when the marked corresponding angles are congruent. P COTheoremTheoremRRESPONDING 3.5 3.5Corresponding Corresponding ANGLES Angles Angles Converse CONVERSE Converse m CorrespondingP Angles Constructing TheoremP Parallel Lines P Have students work in pairs to perform a point that is parallel to a given line, ANSWERSANSWERS If twoIf two lines lines are cutare bycut a by transversal a transversal so the so theD A D A (Thm. 3.1). A the construction(3x + 5)° =SOLUTION 65 Constructing° Use the Parallel Corresponding Angles Converse to write an equation. students should take turns reading 1.1. yes;yes; TheThe angleangle that is corresponding m Usecorresponding a correspondingcompass andm angles straightedge angles are congruent,are tocongruent, construct then thena line m 2 P 2 m Q throughQ B point P that is parallel to line m. Q B Q B Lines. As they3x = construct 60 aSubtract line through 5 from each side. the instructions aloud and executing withwith thethe 7575°° angleangle also formsforms aa the thelines lines are areparallel. parallel. j Stepj 1 Step 2 Step 3 Step 4 6 6 m a point that is parallel to a given line, the steps. linearlinear pairpair withwith the 105°° angle.angle. So,So, k x = 20 Divide each side by 3. Draw a point and line DrawSOLUTION arcsProof Draw Ex. 36, anp. 180arc Copy angle Draw an Draw parallelk lines students should takeC turns reading C C it must be 180° − 105° = 75° by Proof Ex. 36, p. 180 j ʈ k P So, lines m andP n are parallel when x = 20. P P it must be 180° − 105° = 75° by Start by drawing point P with center Q that crosses arc with radius AB and j ʈ k the instructions aloud and executing D the Linear Pair Postulate (Post. 2.8). Step 1 Step 2 Step 3 Step 4 Draw ⃖PD⃗ . This line is A A D A the Linear Pair Postulate (Post. 2.8). and line m. Choose a center A. Using the same m m m m Because the corresponding angles ⃖QP⃗ and line m. Label parallel to line mQ . the steps. Q B Q B Q B point Q anywhere on Using thecompass Corresponding setting, drawAngles an Converse Help in English and Spanish at BigIdeasMath.com Becausehave the thesame corresponding measure, they angles are points1, A and C B. Using the C C Monitoring2. Progress P P Using theP Corresponding AnglesP Converse line m and draw QP⃖⃗ . same compass setting, arc with center C. Label Is there enough information in the diagram to conclude that m n ? Explain. havecongruent the same by de measure,f nition. theySo, m are ʈ n by Find the value of x that makes m ʈ n . D A D Draw a point and line 1. Draw arcs Draw an arc Copy angle Draw an ʈ Draw parallel lines A A the intersection D. (3x + 5)° m draw anFind arc the with valuem center of x that P. makes m n . m m Start by drawing point P with center Q that crosses arc with radius AB and Draw PD. This line is congruentthe Corresponding by def nition. Angles So, Converse m ʈ n by Q Q B Q B ʈ Q B m ⃖⃗ Label point C. (3x + 5)° and line m. Choose a ⃖QP⃗ and line m. Label center A. Using the same parallel to line m. (Thm. 3.5). 65° 75° the Corresponding Angles Converse m point Q anywhere on points A and B. Using the compass setting, draw an n m 2. (Thm.The hypothesis 3.5). and conclusion of Draw a point and line Draw arcs Draw an arc Copy angle Draw an Draw65 parallel° lines line m and draw QP⃖⃗ . same compass setting, arc with center C. Label Start by drawing point P with center Q that crosses arc with radius AB and the intersection D. the Corresponding Angles Converse SOLUTION Draw ⃖PD⃗ . This line isn draw an arc with center P. 105° n 2. The hypothesis and conclusion of and line m. Choose a ⃖QP⃗ and line m. Label center A. Using the same (Thm. 3.5) are the reverse of the Theorems parallel to line m. Label point C. the Corresponding Angles Converse point Q anywhere on pointsLines ASOLUTION and m Band. Using n are the parallel whencompass the setting,marked drawcorresponding an angles are congruent. Corresponding Angles Theorem line m and draw QP⃖⃗ . same compass Theorem setting, 3.6 Alternatearc with center CInterior. Label Angles Converse (3x + 5)° = 65° Use the Corresponding Angles Converse to write an equation. 2. Explain why the Corresponding Angles Converse is the converse of the (Thm.(Thm. 3.5)3.1). are the reverse of the draw anLines arc with m andcenter n are P. parallelthe intersectionwhen the marked D. corresponding angles are congruent. If two lines are cut by a transversal so the CorrespondingTheorems Angles Theorem (Theorem 3.1). Corresponding Angles Theorem Label point C. 3x = 60 Subtract 5 from each side. alternate(3x + 5) interior° = 65° angles Useare thecongruent, Corresponding then Angles Converse to write an equation. Theorem 3.6 Alternate Interior Angles Converse (Thm. 3.1). j the linesx = are 20 parallel.Divide each side by 3. 138 Chapter5 3 4 Parallel and Perpendicular IfLines two lines are cut by a transversal so the 3x = 60 Subtract 5 from each side. alternate interior angles are congruent, then Theorems So, lines m and n are parallel when x = 20. k the lines are parallel. 5 4 j Theorem 3.6x = 20Alternate InteriorDivide each Angles side by 3.Converse Proof Example 2, p. 140 j ʈ k k If two lines are cut by a transversal so the ALTERNATE So, lines m and n INTERIORare parallelHelp when in xEnglish ANGLES 20. and Spanishhs_geo_pe_0303.indd CONVERSE at BigIdeasMath.com 138 ALTERNATEProof Example EXTERIOR 2, p. 140 ANGLES CONVERSEj ʈ k 1/19/15 9:24 AM Monitoringalternate interior Progress angles are congruent, then = j 1. theTheoremIs therelines areenough parallel. 3.7 information Alternate in the diagram Exterior to conclude Angles thatLaurie’s5 m Converseʈ n ? 4Explain. Notes Teacher Actions k 1 Theorem 3.7 Alternate Exterior Angles Converse If two lines are cut by a transversalHelp inso English the and• Spanish The converse at BigIdeasMath.com of the Corresponding Angles Theorem (Thm. 3.1) from the previous lesson is also 1 MonitoringProof Example 2,Progress p. 140 j ʈ k If two lines are cut by a transversal so the alternate exterior angles are congruent,75° then j a theorem, the Corresponding Angles Conversealternate (Thm. exterior3.5). The angles converses are congruent, of the then other angle j 1.the Is lines there areenough parallel. information in the diagram to mconclude that m ʈ n ? Explain. k the lines are parallel. Theorem 3.7 Alternate Exterior Angles Conversetheorems are also8 theorems and are proven to be true using the Corresponding Angles Converse.8 k 105° n 1 If two lines are cut by a transversal so the Turn and Talk: “Explain what it means to findProof a value Ex. 11, of p. x 142 that makes m ʈ n.” There is Proof Ex. 11, p. 142 75 j ʈ k j ʈ k alternate exterior angles are congruent, then ° one value jof x that makes the expression 3x + 5 equal to 65. If the corresponding angles are the lines are parallel. m 2. Explain why the Corresponding Angles Converse is the conversecongruent,8 of the k then m ʈ n by the Corresponding AnglesTheorem Converse. 3.8 Consecutive Interior Angles Converse TheoremCorresponding 3.8 Angles Consecutive Theorem (Theorem Interior 3.1). Angles Converse Proof Ex. 11, p. 142 105° n j ʈ k If two lines are cut by a transversal so the CONSECUTIVEIf two lines are cut by INTERIOR a transversal so theANGLES CONVERSE consecutive interior angles are supplementary, 138 Chapter 3 Parallel and Perpendicularconsecutive Lines interior angles are supplementary, then the lines are parallel. 3 Theorem 3.8 Consecutive Interior Angles Converse j 2. Explain why the Corresponding Angles Converse is the converse of the 5 then the lines are parallel. 3 j If twoCorresponding lines are cut by Angles a transversal Theorem so the(Theorem 3.1). k consecutive interior angles are supplementary, 5 then the lines are parallel. 3 k j If 3 and 5 are hs_geo_pe_0303.indd 138 5 1/19/15 9:24 AM Proof Ex. 12, p. 142 138 Chapter 3 Parallel and Perpendicular Lines supplementary, then j ʈ k. 138 Chapter 3 k Laurie’s Notes TeacherProof Actions Ex. 12, p. 142 If 3 and 5 are supplementary, then j ʈ k. Section 3.3 Proofs with Parallel Lines 139 • The converse of the CorrespondingProof Angles Ex. Theorem 12, p. 142 (Thm. 3.1) from the previousIf 3 and lesson 5 are is also supplementary, then j ʈ k. a theorem, the Corresponding Angles Converse (Thm. 3.5). The converses of the other angle hs_geo_pe_0303.indd 138 Section 3.3 Proofs with Parallel Lines 1/19/15139 9:24 AM theorems are also theoremshscc_geo_te_0303.indd and are proven 138 to be true usingSection the 3.3 Corresponding Proofs with Angles Parallel Converse. Lines 139 2/12/15 2:04 PM Turn and Talk: “Explain what it means to find a value of x that makes m ʈ n.” Therehs_geo_pe_0303.indd is 139 1/19/15 9:24 AM Laurie’s Notes TeacherProving Actions Theorems about Parallel Lines Extra Example 2 one value of x that makes the expression 3x + 5 equal to 65. If the corresponding angles are • The converse of the CorrespondingPROVING Angles Theorem ALTERNATE (Thm. 3.1) INTERIOR from the previous ANGLES lesson CONVERSE is also Laurie’s Notes Teacher Actions Prove the Alternate Interior Angles congruent, then m ʈ n by the Corresponding Angles Converse. Proving the Alternate Interior Angles Converse hs_geo_pe_0303.indd 139 1/19/15 9:24 AM Converse without using the Verticalhs_geo_pe_0303.indd 139 a theorem, the Corresponding Angles Converse (Thm. 3.5). The converses of the other angle “What evidence1/19/15 9:24is needed AM for you to know that constructed lines are parallel?” Corresponding Prove that if two lines are cut by a transversal so the alternate interior angles are angles will have to be congruent. Angles Congruence Theorem. theoremsLaurie’s are also Notes theorems andTeacher congruent,are proven then Actions to the be lines true are usingparallel. the Corresponding Angles Converse. Laurie’s Turn and Talk: Notes “Explain whatTeacher it means to find Actions a value of x that makes m ʈ n.” There is • Make a note of the names for these theorems. They are all converses of previously proven “What evidence is needed for you to know that constructed lines are parallel?” Corresponding one value of x that makes the expressionSOLUTION 3x + 5 equal to 65. If the corresponding1 angles are theorems. In the original theorems, lines were parallel and we concluded a relationship about “Whatang evidenceles will have is to needed be congruent. for you to know that constructed lines4 are parallel?”g Corresponding Given 4 ≅ 5 angles. In the converses, the angle relationship is known and we conclude the lines are parallel. 3 4 g congruent,• Make then a note m ʈof n the by namesthe Corresponding for these theorems. Angles TheyConverse. are all converses of5 previously proven 5 angles will have to be congruent. theorems. In the original theorems,Prove g ʈlines h were parallel and we concluded a relationshiph about 138 Chapter 73 h • Make a note of the names for these theorems. They are all converses of previously proven angles. In the converses, the angle relationship is known and we conclude the lines are parallel. theorems. In the original theorems,STATEMENTS lines were REASONS parallel and we concluded a relationship about Given 4 ≅ 5 angles. In the converses, the1. angle4 5 relationship1. Given is known and we conclude the lines are parallel. Prove g ʈ h ≅ Statements (Reasons) 2. 1 ≅ 4 2. Vertical Angles Congruence Theorem (Theorem 2.6) hscc_geo_te_0303.indd 138 2/12/15 2:04 PM 1. 4 ≅ 5 (Given) 3. 1 ≅ 5 3. Transitive Property of Congruence (Theorem 2.2) Section 3.3 139 1382. 4 is supplementary Chapter 3 to 3. 4. g ʈ h 4. Corresponding Angles Converse
(Linear Pair Post. 2.8)
3. 5 is supplementary to 7. Determining Whether Lines Are Parallel Section 3.3 139 (Linear Pair Post. 2.8) HW3.3: 3-8all, 13-18all, 21-24all, 27,28 rs In the diagram, r ʈ s and 1 is congruent to 3. Prove p ʈ q. 4. 3 ≅ 7 (Congr. Suppl. Thm. 2.4) Fill Up Proofs (AlA, SSIA,hscc_geo_te_0303.indd AEA, SSEA) 139; 2/12/15 2:04 PM hscc_geo_te_0303.indd 138 SOLUTION 2/12/15 2:04 PM 5. g ʈ h (Corr. ⦞ Converse 3.5) p 3 Look at the diagram Redo to make Constructions a plan. The diagram suggests (to thatbe youcollected) look at ; Perpendicular Bisector, PerpendicularSection Line 3.3 from an outside 139 hscc_geo_te_0303.indd 139 12 angles 1, 2, point,and 3. Also, Perpendicular you may f nd it helpful toline focus onfrom one pair a of pointlines on the line, Parallel Line 2/12/15 2:04 PM Extra Example 3 q and one transversal at a time. In the diagram, p ʈ q and 1 is Plan for Proof a. Look at 1 and 2. 1 ≅ 2 because r ʈ s. supplementary to 2. Prove r ʈ s using a b. Look at 2 and 3. If 2 ≅ 3, then p ʈ q. paragraph proof. Plan for Action a. It is given that r ʈ s, so by the Corresponding Angles Theorem hscc_geo_te_0303.indd 139 (Theorem 3.1), 1 ≅ 2. 2/12/15 2:04 PM rs b. It is also given that 1 ≅ 3. Then 2 ≅ 3 by the Transitive Property of Congruence (Theorem 2.2). So, by the Alternate Interior Angles Converse, p q. 3 1 p ʈ
2 Monitoring Progress Help in English and Spanish at BigIdeasMath.com q 3. If you use the diagram below to prove the Alternate Exterior Angles Converse, what Given and Prove statements would you use?
It is given that 1 is supplementary to 1 2. So m 1 + m∠ 2 = 180° by the j definition of supp. ⦞ . It is given that 8 k p ʈ q, so by the Alternate Interior Angles Theorem (Thm.3.2), 2 ≅ 3. By the 4. Copy and complete the following paragraph proof of the Alternate Interior Angles Converse using the diagram in Example 2. def. of ≅ ⦞ , m 2 = m 3. By the It is given that 4 ≅ 5. By the ______, 1 ≅ 4. Then by the Transitive Substitution Prop., m 1 + m 3 = 180°. Property of Congruence (Theorem 2.2), ______. So, by the ______, g ʈ h. So 1 and 3 are supplementary by the def. of supp. ⦞ . Then, by the Consecutive 140 Chapter 3 Parallel and Perpendicular Lines Interior Angles Converse (Thm. 3.8), r ʈ s.
MONITORING PROGRESS hs_geo_pe_0303.indd 140 1/19/15 9:24 AM ANSWERS 3. Given: 1 ≅ 8 Laurie’s Notes Teacher Actions Prove: j ʈ k • MP3: To prove the Alternate Interior Angles Converse (Thm. 3.6), the only method available 4. Vertical Angles Congruence Theorem is to use the Corresponding Angles Converse (Thm 3.5). Once proven, there are two methods (Thm. 2.6); 1 ≅ 5; Corresponding available to use as justification. Proofs of the remaining theorems are left for the exercise set. Angles Converse (Thm. 3.5) • In Example 3, labeling 2 is a hint that you may want to eliminate for some students. Teaching Tip: Cover the top part of the diagram so that line p is not seen. “What do you know about 1 when r ʈ s?” 1 ≅ 2 • Think-Pair-Share: Have students work on Monitoring Progress and then check with their partners.
140 Chapter 3
hscc_geo_te_0303.indd 140 2/12/15 2:04 PM