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Theorems About Parallel Lines

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Theorems About Parallel Lines 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 TeachingTeaching 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 parallel lines Prove— theorems about parallel lines. spherical geometry, all points are points on the surface Notebook Development 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, Previous — supplementary angles 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. m2 = 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 angleslike theare one supplementary. shown. Proof Example 4, p. 134 12(Thm. 2.6), m2 = 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 m6 = 75°. By the Corresponding Angles to solve Example 1. Another way is toTheorem use the 3.3 Alternate Exterior AnglesA Theorem Theorem (Thm. 3.1), m7 = 75°. __________________________________________________________________________________ Corresponding Angles 1 2 1 B Identifying Angles — Theorem to f nd m5 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 angles are congruent.120°. Identify the angles. Explain your reasoning. WARM UP: Theorem (Theorem 2.6) 120º 52 6 E Mini-Assessment a. The measure of 1 is 70 . Find m 2 and m 3. to f nd m4 and m8. ° 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, m8 = 120°. DISCUSSION c. If m1 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 mopening5 = 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 m7 = 72°, find m2 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 m5. 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), m2 = 75°. By the Alternate m5 = 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 m8 = 115° and Interior Angles Theorem (Thm. 3.2), Probing Question: “Will there136 always Chapter be four 3 acuteParallel 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 m6 = 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), m7 = 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 m5 m1 = 42°. What is m2? 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 m4 and m8. Same Side Exterior• AnglePractice 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, m8 = 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), m5 = 120°.
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