4.2 Transversals and Parallel Lines
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1 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? Exlore G.5.A Invesigae aerns o make conjecures abou geomeric relaionshis, including angles formed by arallel lines cu by a ransversal.... Also G.6.A 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 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 is arallel o line, wrien ǁ. The arrows on he lines also indicae ha hey are arallel. Choose aroriae ools such as geomery sofware or a comass, sraighedge, and roracor o exlore he angles formed when a ransversal inersecs arallel lines. Examle 1 and 5 3 and 6 3 and 5 1 and 7 Use your ools o consruc arallel lines and a ransversal ha is no erendicular. Label he angles wih numbers as shown in he diagram. Module Lesson 2
2 B C Measure he angles formed by he arallel lines and he ransversal. Wrie he angle measures for your Figure 1 from se A in he able below. Use your ools o consruc anoher se of arallel lines and a differen ransversal. Label he angles as you did in se A. Measure he angles. Wrie he angle measures for your Figure 2 in he able below. Angle Figure 1 Figure 2 D Look for a Paern Idenify he airs of angles in he diagram. Look for a aern o make conjecures abou heir measures. corresonding angles same-side inerior angles alernae inerior angles alernae exerior angles 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. Reflec 1. Exlain how you can find m 3 in he diagram if m n and m 5 = Wha If? If m n, how many airs of same-side inerior angles are shown in he figure? Wha are he airs? m n 7 Houghon Mifflin Harcour Publishing Comany Module Lesson 2
3 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 Prove he Alernae Inerior Angles Theorem. Given: 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 Subsiuion Proery of Eualiy Linear Pair Theorem 1. Saemens Reasons 2. 3 and 6 are sulemenary. 3. m 3 + m 6 = 180 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 3. In he figure, exlain why 1, 3, 5, and 7 all have he same measure. Module Lesson 2
4 4. Suose m 4 = 57 in he figure shown. Describe wo differen ways o deermine m 6. 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 Angles Theorem. Given: Prove: m 4 = m 8 By he given saemen,. 4 and 6 form a air of. So, using he Alernae Inerior Angles Theorem,. 6 and 8 form a air of verical angles. So, using he Verical Angles Theorem,. Using he in m 4 = m 6, subsiue for m 6. The resul is. 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. 6. Suose m 4 = 36. Find m 5. Exlain. Houghon Mifflin Harcour Publishing Comany Module Lesson 2
5 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. In he diagram, roads a and b are arallel. Exlain how o find he measure of VTU. a b (x + 40) P Q T R V S U (2x - 22) 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 Angles Theorem. 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. B In he diagram, roads a and b are arallel. Exlain how o find he measure of m WUV. a T b U P Q V (9x) S R W (22x + 25) I is given ha m PRS = (9x) and m WUV = (22x + 25). m PRS = m RUW Houghon Mifflin Harcour Publishing Comany by he. RUW and are sulemenary angles. So, m RUW + m WUV =. Solving for x, 31x + 25 = 180, and.subsiue he value of x o find ; m WUV = (22(5) + 25). Module Lesson 2
6 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. 126 (12x + 2y) 36 (3x + 2y) 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)? 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. 10. Essenial Quesion Check-In Why is i imoran o esablish he Same-Side Inerior Angles Posulae before roving he oher heorems? Houghon Mifflin Harcour Publishing Comany Module Lesson 2
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. Online Homework Hins and Hel Exra Pracice m n A. 4 and 6 Corresonding Angles B. 5 and 8 Same-Side Inerior Angles C. 2 and 6 Alernae Inerior Angles D. 4 and 5 Verical Angles 2. Comlee he definiion: A is a line ha inersecs wo colanar lines a wo differen oins. Use he figure o find angle measures. In he figure,. Houghon Mifflin Harcour Publishing Comany 3. Suose m 4 = 82. Find m Suose m 3 = 105. Find m Suose m 3 = 122. Find m Suose m 4 = 76. Find m Suose m 5 = 109. Find m Suose m 6 = 74. Find m 2. Module Lesson 2
8 Use he figure o find angle measures. In he figure, m n and x y. m n x y 9. Suose m 5 = 69. Find m Suose m 9 = 115. Find m Suose m 12 = 118. Find m Suose m 4 = 72. Find m Suose m 4 = 114. Find m Suose m 5 = 86. Find m 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 Houghon Mifflin Harcour Publishing Comany Module Lesson 2
9 In he diagram of movie heaer seas, he incline of he floor, ƒ, is arallel o he seas, s. s f 1 (5y - 7) 3x 16. If m 1 = 60, wha is x? 17. If m 1 = 68, wha is y? 18. Comlee a roof in aragrah form for he Alernae Inerior Angles Theorem. Given: Houghon Mifflin Harcour Publishing Comany Prove: m 3 = m 5 I is given ha, so by he Same-Side Inerior Angles Posulae, 3 and 6 are. When angles are sulemenary, he sum of heir measures is. You can wrie his as m 3 + m 6 = 180. When you look a he given diagram, you see ha 5 and 6 form a line, and so hey are a, which makes hem. You can wrie his as m 5 + m 6 = 180. Using he Subsiuion Proery of Eualiy, you can subsiue 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 of Eualiy, you can subrac from boh sides. So,. Module Lesson 2
10 19. Wrie a roof in wo-column form for he Corresonding Angles Theorem. Given: Prove: m 1 = m 5 Saemens Reasons 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. Houghon Mifflin Harcour Publishing Comany Module Lesson 2
11 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 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. A 50 B F G C 82 r 30 H J D (3n + 7) 22. Which is a rue saemen abou he measure of DCJ? A. I is 30, by he Alernae Inerior Angles Theorem. Houghon Mifflin Harcour Publishing Comany B. I is 30, by he Corresonding Angles Theorem. C. I is 50, by he Alernae Inerior Angles Theorem. D. I is 50, by he Corresonding Angles Theorem. 23. Which is a rue saemen abou he value of n? A. I is 25, by he Alernae Inerior Angles Theorem. B. I is 25, by he Same-Side Inerior Angles Posulae. C. I is 35, by Alernae Inerior Angles Theorem. D. I is 35, by he Corresonding Angles Theorem. Module Lesson 2
12 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. Washingon Sree Aex Comany 51 x Lincoln Sree 37 a. Find x, he measure of he angle formed by he wo roads. Exlain how you found x. 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. Houghon Mifflin Harcour Publishing Comany Module Lesson 2
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