Transversals and Parallel Lines

Transcription

1 COMMON CORE l m Locker LESSON 4.2 Transversals and Parallel Lines Name Class Dae 4.2 Transversals and Parallel Lines Essenial Quesion: How can you rove and use heorems abou angles formed by ransversals ha inersec arallel lines? Common Core Mah Sandards The suden is execed o: COMMON CORE G-CO.C.9 Prove heorems abou lines and angles. Exlore Exloring Parallel Lines and Transversals 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. Resource Locker Mahemaical Pracices COMMON CORE MP.3 Logic Language Objecive Exlain o a arner how o idenify he angles formed by wo arallel lines cu by a ransversal. ENGAGE Essenial Quesion: How can you rove and use heorems abou angles formed by ransversals ha inersec arallel lines? Possible answer: Sar by esablishing a osulae abou cerain airs of angles, such as same-side inerior angles. The osulae allows you o rove a heorem abou oher airs of angles, such as alernae inerior angles. You can hen use he osulae and he heorem o rove oher heorems abou oher airs of angles, such as corresonding angles. Houghon Mifflin Harcour Publishing Comany Image Credis: Ruud Morijn Phoograher/Shuersock Angle Pair Corresonding angles lie on he same side of he ransversal and on he same sides of he inerseced lines. Same-side inerior angles lie on he same side of he ransversal and beween he inerseced lines. Alernae inerior angles are nonadjacen angles ha lie on oosie sides of he ransversal beween he inerseced lines. Alernae exerior angles lie on oosie sides of he ransversal and ouside he inerseced lines. Recall ha arallel lines lie in he same lane and never inersec. In he figure, line l is arallel o line m, wrien lǁm. The arrows on he lines also indicae ha hey are arallel. l m l m Examle 1 and 5 3 and 6 3 and 5 1 and 7 PREVIEW: LESSON PERFORMANCE TASK View he Engage secion online. Discuss he hoo. Ask sudens o describe ways ha a real-world examle of arallel lines and a ransversal like his examle differ from arallel lines and a ransversal ha migh aear in a geomery book. Then review he Lesson Performance Task. Module Lesson 2 Name Class Dae 4.2 Transversals and Parallel Lines Essenial Quesion: How can you rove and use heorems abou angles formed by ransversals ha inersec arallel lines? G-CO.C.9 Prove heorems abou lines and angles. Houghon Mifflin Harcour Publishing Comany Image Credis: Ruud Morijn Phoograher/Shuersock Exlore Exloring Parallel Lines and Transversals 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. Angle Pair Examle Corresonding angles lie on he same side of he ransversal and on he same sides of he inerseced lines. Same-side inerior angles lie on he same side of he ransversal and beween he inerseced lines. Alernae inerior angles are nonadjacen angles ha lie on oosie sides of he ransversal beween he inerseced lines. Alernae exerior angles lie on oosie sides of he ransversal and ouside he inerseced lines. Recall ha arallel lines lie in he same lane and never inersec. In he figure, line l is arallel o line m, wrien lǁm. The arrows on he lines also indicae ha hey are arallel. l m Resource 1 and 5 3 and 6 3 and 5 1 and 7 HARDCOVER PAGES Turn o hese ages o find his lesson in he hardcover suden ediion. Module Lesson Lesson 4.2

2 When arallel lines are cu by a ransversal, he angle airs formed are eiher congruen or sulemenary. The following osulae is he saring oin for roving heorems abou arallel lines ha are inerseced by a ransversal. Same-Side Inerior Angles Posulae If wo arallel lines are cu by a ransversal, hen he airs of same-side inerior angles are sulemenary. Follow he ses o illusrae he osulae and use i o find angle measures. A B C D Draw wo arallel lines and a ransversal, and number he angles formed from 1 o 8. Idenify he airs of same-side inerior angles. 4 and 5; 3 and 6 Wha does he osulae ell you abou hese same-side inerior angle airs? Given, hen 4 and 5 are sulemenary and 3 and 6 are sulemenary. If m 4 = 70, wha is m 5? Exlain. m 5 = 110 ; 4 and 5 are sulemenary, so m 4 + m 5 = 180. Therefore 70 + m 5 = 180, so m 5 = 110. Reflec 1. Exlain how you can find m 3 in he diagram if and m 6 = and 6 are sulemenary, so m 3 + m 6 = 180. Therefore m = 180, so m 3 = Wha If? If m n, how many airs of same-side inerior angles are shown in he figure? Wha are he airs? Two airs; 3 and 5, 4 and 6 m n 7 Houghon Mifflin Harcour Publishing Comany EXPLORE Exloring Parallel Lines and Transversals INTEGRATE TECHNOLOGY The roeries of arallel lines and ransversals can be exlored using geomery sofware. Sudens can dislay lines and angle measures on screen, roae a line so i is arallel o anoher, and observe he relaionshis beween angles. QUESTIONING STRATEGIES When wo lines are cu by a ransversal, how many airs of corresonding angles are formed? How many airs of same-side inerior angles? 4; 2 Wha does he Same-Side Inerior Angles Posulae ell you abou he measure of a air of same-side inerior angles? The sum of heir measures is 180. Module Lesson 2 PROFESSIONAL DEVELOPMENT Mah Background When wo arallel lines are cu by a ransversal, alernae inerior angles have he same measure, corresonding angles have he same measure, and same-side inerior angles are sulemenary. One of hese hree facs mus be aken as a osulae and hen he oher wo may be roved. In he work ex, he saemen abou same-side inerior angles is aken as he osulae. This is closely relaed o one of he osulaes saed in Euclid s Elemens: If a sraigh line falling on wo sraigh lines makes he inerior angles on he same side less han wo righ angles, he wo sraigh lines, if roduced indefiniely, mee on ha side on which are he angles less han he wo righ angles. Transversals and Parallel lines 176

3 EXPLAIN 1 Proving ha Alernae Inerior Angles are Congruen CONNECT VOCABULARY Have sudens exerience he idea of wha he erm alernae means by shading in alernaing figures in a series of figures. QUESTIONING STRATEGIES Do you have o wrie a searae roof for every air of alernae inerior angles in he figure? Why or why no? No, you can wrie he roof for any air of alernae inerior angles. The same reasoning alies o all he airs. Exlain 1 Proving ha Alernae Inerior Angles are Congruen Oher airs of angles formed by arallel lines cu by a ransversal are alernae inerior angles. Alernae Inerior Angles Theorem If wo arallel lines are cu by a ransversal, hen he airs of alernae inerior angles have he same measure. To rove somehing o be rue, you use definiions, roeries, osulaes, and heorems ha you already know. Examle 1 Given: Prove he Alernae Inerior Prove: m 3 = m 5 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 Subracion Proery of Eualiy Given Definiion of sulemenary angles 1. Saemens 2. 3 and 6 are sulemenary. Subsiuion Proery of Eualiy Linear Pair Theorem 1. Given Reasons 2. Same-Side Inerior Angles Posulae 3. m 3 + m 6 = Definiion of sulemenary angles Houghon Mifflin Harcour Publishing Comany 4. 5 and 6 are a linear air and 6 are sulemenary. 6. m 5 + m 6 = m 3 + m 6 = m 5 + m 6 8. m 3 = m 5 Reflec 4. Given 5. Linear Pair Theorem 6. Definiion of sulemenary angles 7. Subsiuion Proery of Eualiy 8. Subracion Proery of Eualiy 3. In he figure, exlain why 1, 3, 5, and 7 all have he same measure. m 1 = m 3 and m 5 = m 7 (Verical Angles Theorem), m 3 = m 5 (Alernae Inerior Angles Theorem), so m 1 = m 3 = m 5 = m 7 (Transiive Proery of Eualiy). Module Lesson 2 COLLABORATIVE LEARNING Peer-o-Peer Aciviy Have sudens use lined aer or geomery sofware o draw wo arallel lines and a ransversal ha is no erendicular o he lines. Insruc he suden s arner o shade or mark he acue angles wih one color and he obuse angles wih anoher color. Le sudens use a roracor or geomery sofware o see ha all he angles wih he same color are congruen, and ha airs of angles wih differen colors are sulemenary. 177 Lesson 4.2

4 4. Suose m 4 = 57 in he figure shown. Describe wo differen ways o deermine m 6. By he Alernae Inerior Angles Theorem, m 6 = 57. Also 4 and 5 are sulemenary, so m 5 = 123. Since 5 and 6 are sulemenary, m 6 = 57. Exlain 2 Proving ha Corresonding Angles are Congruen Two arallel lines cu by a ransversal also form angle airs called corresonding angles. Corresonding Angles Theorem If wo arallel lines are cu by a ransversal, hen he airs of corresonding angles have he same measure. Examle 2 Comlee a roof in aragrah form for he Corresonding Given: Prove: m 4 = m 8 EXPLAIN 2 Proving ha Corresonding Angles are Congruen Focus on Reasoning MP.2 Exlaining and jusifying argumens is a he hear of his roof-based lesson. You may wish o have sudens air u wih a roof buddy. Sudens can exchange heir work wih his arner and check ha he arner s logical argumens make sense. By he given saemen,. 4 and 6 form a air of alernae inerior angles. So, using he Alernae Inerior Angles Theorem, m 4 = m 6. 6 and 8 form a air of verical angles. So, using he Verical Angles Theorem, m 6 = m 8. Using he Subsiuion Proery of Eualiy in m 4 = m 6, subsiue m 4 for m 6. The resul is m 4 = m 8. Reflec 5. Use he diagram in Examle 2 o exlain how you can rove he Corresonding Angles Theorem using he Same-Side Inerior Angles Posulae and a linear air of angles. By he Same-Side Inerior Angles Theorem, m 4 + m 5 = 180. As a linear air, m 4 + m 1 = 180. Therefore m 4 + m 1 = m 4 + m 5, so m 1 = m Suose m 4 = 36. Find m 5. Exlain. m 5 = 144 ; Since 1 and 4 form a linear air, hey are sulemenary. So, m 1 = 144. Using he Corresonding Angles Theorem, you know ha m 1 = m 5. So, m 5 = 144. Module Lesson 2 Houghon Mifflin Harcour Publishing Comany QUESTIONING STRATEGIES Wha can you use as reasons in a roof? given informaion, roeries, osulaes, and reviously-roven heorems How can you check ha he firs and las saemens in a wo-column roof are correc? The firs saemen should mach he Given and he las saemen should mach he Prove. Focus on Modeling MP.4 Draw arallel lines and a ransversal on a ransarency. Trace an acue and an obuse angle formed by he lines ono anoher ransarency, and use hem o find congruen angles. DIFFERENTIATE INSTRUCTION Kinesheic Exerience Have sudens draw a air of arallel lines and a ransversal on ranslucen aer suares. Tear he aer beween he arallel lines, and overlay he wo ars o show ha he angles are congruen. AVOID COMMON ERRORS Some sudens may have difficuly idenifying he correc angles for wo arallel lines cu by a ransversal because hey are unfamiliar wih he angles. Have hese sudens use highlighers o color-code he differen angle airs. For examle, sudens can use a yellow highligher o highligh he erm corresonding angles and hen use he same highligher o mark he corresonding angles. Transversals and Parallel lines 178

5 EXPLAIN 3 Using Parallel Lines o Find Angle Pair Relaionshis Exlain 3 Using Parallel Lines o Find Angle Pair Relaionshis You can aly he heorems and osulaes abou arallel lines cu by a ransversal o solve roblems. Examle 3 Find each value. Exlain how o find he values using osulaes, heorems, and algebraic reasoning. Focus on Modeling MP.4 Have sudens look hrough magazines o find icures wih arallel lines and ransversals, such as bridges, fences, furniure, ec. Use markers or colored ae o mark he lines, and hen idenify angles ha aear o be congruen and angles ha aear o be sulemenary. QUESTIONING STRATEGIES How can you check ha he osulaes and heorems abou arallel lines aly o realworld siuaions? Samle answer: Measure some angle air relaionshis wih lines ha aear arallel, and verify ha he osulaes and heorems aly. ELABORATE AVOID COMMON ERRORS Sudens may incorrecly aly he osulaes and heorems resened in his lesson when lines cu by a ransversal are no arallel. Remind hem ha he osulaes and heorems are only rue for arallel lines. Houghon Mifflin Harcour Publishing Comany In he diagram, roads a and b are arallel. Exlain how o find he measure of VTU. I is given ha m PRQ = (x + 40) and m VTU = (2x - 22). m PRQ = m RTS by he Corresonding Angles Theorem and m RTS = m VTU by he Verical So, m PRQ = m VTU, and x + 40 = 2x Solving for x, x + 62 = 2x, and x = 62. Subsiue he value of x o find m VTU: m VTU = (2 (62) - 22) = 102. In he diagram, roads a and b are arallel. Exlain how o find he measure of m WUV. I is given ha m PRS = (9x) and m WUV = (22x + 25). (9x) Corresonding Angles Theorem m PRS = m RUW by he. S RUW and WUV are sulemenary angles. So, m RUW + m WUV = 180. Solving for x, 31x + 25 = 180, and x = 5.Subsiue he value of x o find m WUV ; m WUV = (22(5) + 25) = 135. Your Turn 7. In he diagram of a gae, he horizonal bars are arallel and he verical bars are arallel. Find x and y. Name he osulaes and/or heorems ha you used o find he values. (x + 40) P P 126 a R (12x + 2y) x = 10, y = 3; (12x + 2y) = 126 by he Corresonding Angles Theorem Q T R V b a U Q (3x + 2y) S U (2x - 22) T b V (22x + 25) W and (3x + 2y) = 36 by he Alernae Inerior Solving he euaions simulaneously resuls in x = 10 and y = 3. Module Lesson 2 36 QUESTIONING STRATEGIES Posulaes may be used o rove heorems; which osulae was used o rove he wo heorems in his lesson? Same-Side Inerior Angles Posulae LANGUAGE SUPPORT Visual Cues Have sudens work in airs o add he following angle definiions and icures o an organizer like he one below: Angle(s) Definiion Picure Alernae inerior angles Corresonding angles Same-side inerior angles 179 Lesson 4.2

6 Elaborae 8. How is he Same-Side Inerior Angles Posulae differen from he wo heorems in he lesson (Alernae Inerior Angles Theorem and Corresonding Angles Theorem)? The osulae shows ha airs of angles are sulemenary, while he heorems show ha airs of angles have he same measure. SUMMARIZE THE LESSON Have sudens use he diagram o idenify he following: 9. Discussion Look a he figure below. If you know ha and are arallel, and are given one angle measure, can you find all he oher angle measures? Exlain Yes; Possible exlanaion: Consider angles 1 4. If you knew one angle you can use he fac ha angles ha are linear airs are sulemenary and he Verical You could use eiher he Alernae Inerior Angles Theorem or he Corresonding Angles Theorem o find one of he angle measures for angles 5 8. Then you can use he Linear Pair Theorem and he Verical Angles Theorem o find all of hose angle measures. 10. Essenial Quesion Check-In Why is i imoran o esablish he Same-Side Inerior Angles Posulae before roving he oher heorems? You need o use he Same-Side Inerior Angles Posulae o rove he Alernae Inerior Angles Theorem, and o rove he Corresonding Angles Theorem you need o use eiher he Same-SIde Inerior Angles Posulae or he Alernae Inerior Parallel lines and Transversal Congruen angles: Corresonding 1 3; 2 4; 5 7; 6 8 Alernae inerior 2 7; 3 6 Sulemenary angles: Same-side inerior 2 and 3; 6 and 7 Evaluae: Homework and Pracice 1. In he figure below, m n. Mach he angle airs wih he correc label for he airs. Indicae a mach by wriing he leer for he angle airs on he line in fron of he corresonding labels. A. 4 and 6 C Corresonding Angles B. 5 and 8 A Same-Side Inerior Angles C. 2 and 6 D Alernae Inerior Angles D. 4 and 5 B Verical Angles m n 7 Online Homework Hins and Hel Exra Pracice Houghon Mifflin Harcour Publishing Comany EVALUATE ASSIGNMENT GUIDE Conces and Skills Exlore Exloring Parallel Lines and Transversals Pracice Exercises 1 4 Module Lesson 2 Exercise Deh of Knowledge (D.O.K.) 1 1 Recall of Informaion MP.2 Reasoning 2 1 Recall of Informaion MP.3 Logic COMMON CORE Mahemaical Pracices Examle 1 Proving ha Alernae Inerior Angles are Congruen Examle 2 Proving ha Corresonding Angles are Congruen Examle 3 Using Parallel Lines o Find Angle Pair Relaionshis Exercises 5 14 Exercises 5 14 Exercises Recall of Informaion MP.2 Reasoning Skills/Conces MP.2 Reasoning 15 3 Sraegic Thinking MP.4 Modeling 16 2 Skills/Conces MP.4 Modeling 17 2 Skills/Conces MP.3 Logic Transversals and Parallel lines 180

7 AVOID COMMON ERRORS Sudens may incorrecly aly he osulaes and heorems resened in his lesson when lines cu by a ransversal are no arallel. Remind hem ha he osulaes and heorems are only rue for arallel lines. Focus on Modeling MP.4 Some sudens may have difficuly idenifying he correc angles for wo arallel lines cu by a ransversal because some combinaions of angles can be visually disracing. Sugges ha hese sudens re-draw or race he diagram for each exercise, labeling only he angles necessary for he exercise. 2. Comlee he definiion: A ransversal is a line ha inersecs wo colanar lines a wo differen oins. Use he figure o find angle measures. In he figure,. 3. Suose m 4 = 82. Find m Suose m 3 = 105. Find m 6. m 5 = 98, by he Same-Side Inerior Angles Posulae. 5. Suose m 3 = 122. Find m Suose m 4 = 76. Find m 6. m 5 = 122, by he Alernae Inerior 7. Suose m 5 = 109. Find m Suose m 6 = 74. Find m 2. m 1 = 109, by he Corresonding Use he figure o find angle measures. In he figure, m n and x y. m 6 = 75, by he Same-Side Inerior Angles Posulae. m 6 = 76, by he Alernae Inerior m 2 = 74, by he Corresonding m n x y Houghon Mifflin Harcour Publishing Comany 9. Suose m 5 = 69. Find m Suose m 9 = 115. Find m 6. m 10 = 69, by he Alernae Inerior m 6 = 115, by he Alernae Inerior 11. Suose m 12 = 118. Find m Suose m 4 = 72. Find m 11. m 7 = 118, by he Alernae Inerior m 11 = 108, by he Corresonding Angles Theorem and Linear Pair Theorem. 13. Suose m 4 = 114. Find m Suose m 5 = 86. Find m 12. m 14 = 66, by he Corresonding Angles Theorem, Linear Pair Theorem, and Corresonding m 12 = 86, by he Alernae Inerior Angles Theorem and Corresonding Angles Theorem. Module Lesson 2 Exercise Deh of Knowledge (D.O.K.) COMMON CORE Mahemaical Pracices Sraegic Thinking MP.3 Logic Sraegic Thinking MP.3 Logic 181 Lesson 4.2

8 15. Ocean waves move in arallel lines oward he shore. The figure shows he ah ha a windsurfer akes across several waves. For his exercise, hink of he windsurfer s wake as a line. If m 1 = (2x + 2y) and m 2 = (2x + y), find x and y. Exlain your reasoning. x = 15 and y = 40; m 2 = 70 by he Corresonding Angles Theorem and m 1 + m 2 = 180 by he Same-Side Inerior Angles Posulae. So, m 1 = = 110. m 2 = (2x + y) and m 2 = 70, so (2x + y) = 70 by he Subsiuion Proery of Eualiy and 2x = 70 - y. m 1 = (2x + 2y), so m 1 = (70 - y + 2y) = (70 + y), so (70 + y) º = 110º and y = 40 by he Subsiuion Proery of Eualiy. Since 2x + y = 70 and y = 40, 2x = 30 and x = 15. In he diagram of movie heaer seas, he incline of he floor, ƒ, is arallel o he seas, s. 16. If m 1 = 60, wha is x? x = 40; by he Corr. s Thm. and he Lin. Pair Thm., 3x + m 1 = 180, so 3x + 60 = 180, and x = If m 1 = 68, wha is y? y = 15; by he Al. In. s Thm., m 1 = (5y - 7), so 68 = 5y - 7, and y = 15. s f (5y - 7) 2 3x PEER-TO-PEER DISCUSSION Ask sudens o discuss wih a arner he arallel line diagrams for various airs of angles inroduced in his lesson. Ask hem o wrie and solve a word roblem abou arallel lines and heir associaed angles on index cards, and swich wih heir arners o solve he roblem. Focus on Mah Connecions MP.1 Remind sudens abou how o use he Same-Side Inerior Angles Posulae and he heorems abou arallel lines o wrie euaions ha will hel hem find angle measures relaed o arallel lines. 18. Comlee a roof in aragrah form for he Alernae Inerior Angles Theorem. Given: Prove: m 3 = m 5 I is given ha, so using he Same-Side Inerior Angles Posulae, 3 and 6 are sulemenary. So, he sum of heir measures is 180 and m 3 + m 6 = 180. You can see from he diagram ha 5 and 6 form a line, so hey are a linear air, which makes hem sulemenary. Then m 5 + m 6 = 180. Using he Subsiuion Proery of Eualiy, you can subsiue 180 in m 3 + m 6 = 180 wih m 5 + m 6. This resuls in m 3 + m 6 = m 5 + m 6. Using he Subracion Proery m 6 m 3 = m 5 of Eualiy, you can subrac from boh sides. So,. Houghon Mifflin Harcour Publishing Comany Module Lesson 2 Transversals and Parallel lines 182

9 JOURNAL Have sudens lis examles in he real world where hey have seen arallel lines cu by a ransversal. Ask hem o name he airs of angles ha are congruen. 19. Wrie a roof in wo-column form for he Corresonding Given: Prove: m 1 = m 5 Saemens Reasons Given 2. m 3 = m 5 2. Alernae Inerior Angles Theorem 3. m 1 = m 3 3. Verical Angles Theorem 4. m 1 = m 5 4. Subsiuion Proery of Eualiy 20. Exlain he Error Angelina wroe a roof in aragrah form o rove ha he measures of corresonding angles are congruen. Idenify her error, and describe how o fix he error. Angelina s roof: I am given ha. 1 and 4 are sulemenary angles because hey form a linear air, so m 1 + m 4 = and 8 are also sulemenary because of he Same-Side Inerior Angles Posulae, so m 4 + m 8 = 180. You can subsiue m 4 + m 8 for 180 in he firs euaion above. The resul is m 1 + m 4 = m 4 + m 8. Afer subracing m 4 from each side, I see ha 1 and 8 are corresonding angles and m 1 = m 8. 4 and 8 are no same-side inerior angles. 4 and 5 are same-side inerior angles. So, in he aragrah roof, relace 8 wih 5 o see ha m 1 = m 5. Houghon Mifflin Harcour Publishing Comany 21. Counerexamle Ellen hinks ha when wo lines ha are no arallel are cu by a ransversal, he measures of he alernae inerior angles are he same. Wrie a roof o show ha she is correc or use a counerexamle o show ha she is incorrec. A ossible diagram is shown, wih wo nonarallel lines cu by a ransversal. I can measure he angles in my drawing wih a roracor as a counerexamle. 4 and 5 are alernae inerior angles, bu m 4 = 90 and m 5 = 130, so he measures are no he same when he lines are no arallel Module Lesson Lesson 4.2

10 H.O.T. Focus on Higher Order Thinking Analyzing Mahemaical Relaionshis Use he diagram of a saircase railing for Exercises 22 and 23. _ AG _ CJ and _ AD _ FJ. Choose he bes answer. 22. Which is a rue saemen abou he measure of DCJ? A. I is 30, by he Alernae Inerior B. I is 30, by he Corresonding C. I is 50, by he Alernae Inerior D. I is 50, by he Corresonding 23. Which is a rue saemen abou he value of n? A. I is 25, by he Alernae Inerior B. I is 25, by he Same-Side Inerior Angles Posulae. F A 50 B C 82 r 30 G H J D (3n + 7) Focus on Criical Thinking MP.3 Sudens may no have encounered geomery roblems ha can be solved by drawing a line or some oher elemen on a given figure. Sudens may objec ha a roblem of his kind isn fair. Poin ou ha many roblems omi informaion. To solve a roblem, you use roblem-solving skills, working your way logically from he saemen of he roblem o is soluion. This roblem inroduces a new aroach sudens can add o heir arsenal of roblem-solving skills. C. I is 35, by Alernae Inerior D. I is 35, by he Corresonding Lesson Performance Task Washingon Sree is arallel o Lincoln Sree. The Aex Comany s headuarers is locaed beween he srees. From headuarers, a sraigh road leads o Washingon Sree, inersecing i a a 51 angle. Anoher sraigh road leads o Lincoln Sree, inersecing i a a 37 angle. a. Find x. Exlain your mehod. Washingon Sree Aex Comany Lincoln Sree b. Suose ha anoher sraigh road leads from he oosie side of headuarers o Washingon Sree, inersecing i a a y angle, and anoher sraigh road leads from headuarers o Lincoln Sree, inersecing i a a z angle. Find he measure of he angle w formed by he wo roads. Exlain how you found w x 37 a. Draw a line arallel o he wo srees and assing hrough he verex of he angle wih measure x. Because alernae inerior angles formed by arallel lines and a ransversal are congruen, he angle measuring x is divided ino a 51 angle and a 37 angle, so x = = 88. b. Use he mehod from ar a. The o ar of he unknown angle measures y and he boom ar measures z. So, w = y + z. 37 Houghon Mifflin Harcour Publishing Comany Focus on Modeling MP.4 If sudens have difficuly drawing he figure for ar b of he Lesson Performance Task, show hem his figure: Washingon Sree y Aex 51 Comany w x z Lincoln Sree 37 Module Lesson 2 EXTENSION ACTIVITY The easies way o solve he roblems in he Lesson Performance Task is o draw a line arallel o Washingon and Lincoln srees hrough he angle wih is verex a he Aex Comany. Bu wha if here are wo such ossible lines, or even more? Have sudens research and reor on Playfair s Axiom, which saes ha, in a lane, no more han one line can be drawn hrough a given oin ha is arallel o a given line. Sudens should be able o gras his conce inuiively, as a second line hrough a oin would form a nonzero angle wih he arallel line and hus could no also be arallel. Scoring Rubric 2 oins: Suden correcly solves he roblem and exlains his/her reasoning. 1 oin: Suden shows good undersanding of he roblem bu does no fully solve or exlain his/her reasoning. 0 oins: Suden does no demonsrae undersanding of he roblem. Transversals and Parallel lines 184