Name Class Date. Angles Formed by Parallel Lines and Transversals Going Deeper. Introducing Transversals t

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1 Name Class Dae 3-2 Angles Formed by Parallel Lines and Transversals Going Deeer Essenial uesion: How can you rove and use heorems abou angles formed by ransversals ha inersec arallel lines? 1 G-CO.3.9 E n g a g e Inroducing Transversals Recall ha a ransversal is a line ha inersecs wo colanar lines a wo differen oins. In he figure, line is a ransversal. The able summarizes he names of angle airs formed by a ransversal. 4 3 Angle Pair Examle Corresonding angles lie on he same side of he ransversal and on he same sides of he inerseced lines. 1 and Same-side inerior angles lie on he same side of he ransversal and beween he inerseced lines. 3 and Alernae inerior angles are nonadjacen angles ha lie on oosie sides of he ransversal beween he inerseced lines. 3 and Alernae exerior angles are angles ha lie on oosie sides of he ransversal ouside he inerseced lines. 2 and The following osulae is he saring oin for roving heorems abou arallel lines ha are inerseced by a ransversal. Houghon Mifflin Harcour Publishing Comany Same-Side Inerior Angles Posulae If wo arallel lines are cu by a ransversal, hen he airs of same-side inerior angles are sulemenary. Given ǁ, 4 and are sulemenary. Given ǁ, 3 and are sulemenary. 4 3 REFLECT 1a. Exlain how you can find m 3 in he osulae diagram if ǁ and m = 1. 1b. In he osulae diagram, suose ǁ and line is erendicular o line. Can you conclude ha line is erendicular o line? Exlain. Chaer 3 9 Lesson 2

2 2 G-CO.3.9 r o o f Alernae Inerior Angles Theorem If wo arallel lines are cu by a ransversal, hen he airs of alernae inerior angles have he same measure. 4 3 Given: ǁ Prove: m 3 = m Comlee he roof by wriing he missing reasons. Choose from he following reasons. You may use a reason more han once. Same-Side Inerior Angles Posulae Given Definiion of sulemenary angles Subracion Proery of Eualiy Subsiuion Proery of Eualiy Linear Pair Theorem Saemens 1. ǁ and are sulemenary m 3 + m = and are a linear air. 4.. and are sulemenary... m + m = 10.. m 3 + m = m + m.. m 3 = m. REFLECT 2a. Suose m 4 = in he above figure. Describe wo differen ways o deermine m. 2b. In he above figure, exlain why 1, 3,, and all have he same measure. Houghon Mifflin Harcour Publishing Comany 2c. In he above figure, is i ossible for all eigh angles o have he same measure? If so, wha is ha measure? Chaer 3 9 Lesson 2

3 3 G-CO.3.9 r o o f Corresonding Angles Theorem If wo arallel lines are cu by a ransversal, hen he airs of corresonding angles have he same measure. Given: ǁ Prove: m 1 = m Comlee he roof by wriing he missing reasons. 4 3 Saemens 1. ǁ m 3 = m m 1 = m m 1 = m 4. REFLECT 3a. Exlain how you can you rove he Corresonding Angles Theorem using he Same-Side Inerior Angles Posulae and a linear air of angles. Houghon Mifflin Harcour Publishing Comany Many osulaes and heorems are wrien in he form If, hen. The converse of such a saemen has he form If, hen. The converse of a osulae or heorem may or may no be rue. The converse of he Same-Side Inerior Angles Posulae is acceed as rue, and his makes i ossible o rove ha he converses of he revious heorems are rue. Converse of he Same-Side Inerior Angles Posulae If wo lines are cu by a ransversal so ha a air of same-side inerior angles are sulemenary, hen he lines are arallel. Converse of he Alernae Inerior Angles Theorem If wo lines are cu by a ransversal so ha a air of alernae inerior angles have he same measure, hen he lines are arallel. Converse of he Corresonding Angles Theorem If wo lines are cu by a ransversal so ha a air of corresonding angles have he same measure, hen he lines are arallel. Chaer 3 9 Lesson 2

4 A aragrah roof is anoher way of resening a mahemaical argumen. As in a wo-column roof, he argumen mus flow logically and every saemen should have a reason. 4 G-CO.3.9 r o o f Eual-Measure Linear Pair Theorem If wo inersecing lines form a linear air of angles wih eual measures, hen he lines are erendicular. Given: m 1 = m 2 Prove: l m Comlee he following aragrah roof. 1 l 2 m I is given ha 1 and 2 form a linear air. Therefore, 1 and 2 are sulemenary by he. By he definiion of sulemenary angles, m 1 + m 2 = 10. I is also given ha m 1 = m 2. So, m 1 + m 1 = 10 by he. Simlifying gives 2m 1 = 10 and m 1 = 90 by he Division Proery of Eualiy. Therefore, 1 is a righ angle and l m by he. REFLECT 4a. Sae he converse of he Eual-Measure Linear Pair Theorem shown above. Is he converse rue? r a c i c e In Exercises 1 2, comlee each roof by wriing he missing saemens or reasons. 1. If wo arallel lines are cu by a ransversal, hen he airs of alernae exerior angles have he same measure. Given: ǁ Prove: m 1 = m Saemens 1. ǁ Houghon Mifflin Harcour Publishing Comany 2. m 1 = m m = m m 1 = m 4. Chaer 3 9 Lesson 2

5 2. P rove he Converse of he Alernae Inerior Angles Theorem. 1 4 iven: m 3 = m G Prove: ǁ Saemens 1. m 3 = m and are a linear air. 2. Definiion of linear air Linear Pair Theorem 4. m + m = m 3 + m = and are sulemenary... ǁ. m 3. Comlee he aragrah roof. 1 iven: ℓ ǁ m and ǁ G Prove: m 1 = m 2 Houghon Mifflin Harcour Publishing Comany I is given ha ǁ, so m 1 = m 3 by he. I is also given ha ℓ ǁ m, so m 3 = m 2 by he. Therefore, m 1 = m 2 by he. 4. T he figure shows a given line m, a given oin P, and he consrucion of a line ℓ ha is arallel o line m. Exlain why line ℓ is arallel o line m. P m Chaer 3 99 Lesson 2

6 . Can you use he informaion in he figure o conclude ha ǁ? Why or why no? Houghon Mifflin Harcour Publishing Comany Chaer Lesson 2

7 Name Class Dae 3-2 Addiional Pracice Find each angle measure. 1. m 1 2. m 2 3. m ABC 4. m DEF Comlee he wo-column roof o show ha same-side exerior angles are sulemenary.. Given: Prove: m 1 + m 3 = 10 Proof: Saemens Houghon Mifflin Harcour Publishing Comany Given 2. a. 2. Lin. Pair Thm b. 4. c. 4. Def. of s. d.. e.. Ocean waves move in arallel lines oward he shore. The figure shows Sandy Beaches windsurfing across several waves. For his exercise, hink of Sandy s wake as a line. m 1 = (2x + 2y) and m 2 = (2x + y). Find x and y. x = y = Chaer Lesson 2

8 Problem Solving Find each value. Name he osulae or heorem ha you used o find he values. 1. In he diagram of movie heaer seas, 2. In he diagram, roads a and b are arallel. he incline of he floor, f, is arallel o he seas, s. If m 1 =, wha is x? Wha is he measure of PQR? 3. In he diagram of he gae, he horizonal bars are arallel and he verical bars are arallel. Find x and y. Use he diagram of a saircase railing for Exercises 4 and. AG CJ and AD FJ. Choose he bes answer. 4. Which is a rue saemen abou he measure of DCJ? A I euals 30, by he Alernae Inerior Angles Theorem. B I euals 30, by he Corresonding Angles Posulae. C I euals 0, by he Alernae Inerior Angles Theorem. D I euals 0, by he Corresonding Angles Posulae.. Which is a rue saemen abou he value of n? F I euals 2, by he Alernae Inerior Angles Theorem. G I euals 2, by he Same-Side Inerior Angles Theorem. H I euals 3, by he Alernae Inerior Angles Theorem. J I euals 3, by he Same-Side Inerior Angles Theorem. Houghon Mifflin Harcour Publishing Comany Chaer Lesson 2