Explore 2 Proving he Verical Angles Theorem The conjecure from he Explore abou verical angles can be proven so i can be saed as a heorem. The Verical Angles Theorem If wo angles are verical angles, hen he angles are congruen. 1 4 3 2 1 3 and 2 4 You have wrien proofs in wo-column and paragraph proof formas. Anoher ype of proof is called a flow proof. A flow proof uses boxes and arrows o show he srucure of he proof. The seps in a flow proof move from lef o righ or from op o boom, shown by he arrows connecing each box. The jusificaion for each sep is wrien below he box. You can use a flow proof o prove he Verical Angles Theorem. Follow he seps o wrie a Plan for Proof and a flow proof o prove he Verical Angles Theorem. Given: 1 and 3 are verical angles. Prove: 1 3 4 1 3 2 Module 4 164 Lesson 1
A Complee he final seps of a Plan for Proof: Because 1 and 2 are a linear pair and 2 and 3 are a linear pair, hese pairs of angles are supplemenary. This means ha m 1 + m 2 = 180 and m 2 + m 3 = 180. By he Transiive Propery, m 1 + m 2 = m 2 + m 3. Nex: Subrac m 2 from boh sides o conclude ha m 1 = m 3. So, 1 3. B Use he Plan for Proof o complee he flow proof. Begin wih wha you know is rue from he Given or he diagram. Use arrows o show he pah of he reasoning. Fill in he missing saemen or reason in each sep. 1 and 2 are a linear pair. Given (see diagram) 1 and 3 are verical angles. Given 1 and 2 are supplemenary. Linear Pair Theorem m 1 + m 2 = 180 Def. of supplemenary angles 2 and 3 are a linear pair. Given (see diagram) 2 and 3 are supplemenary. Linear Pair Theorem m 2 + m 3 = 180 Def. of supplemenary angles 1 3 m 1 = m 3 m 1 + m 2 = m 2 + m 3 Def. of congruence Subracion Propery of Equaliy Transiive Propery of Equaliy 4. Discussion Using he oher pair of angles in he diagram, 2 and 4, would a proof ha 2 4 also show ha he Verical Angles Theorem is rue? Explain why or why no. Yes, i does no maer which pair of verical angles is used in he proof. Similar saemens and reasons could be used for eiher pair of verical angles. 5. Draw wo inersecing lines o form verical angles. Label your lines and ell which angles are congruen. Measure he angles o check ha hey are congruen. A D E Possible answer: C B By he Verical Angles Theorem, AEC DEB and AED CEB. Checking by measuring, m AEC = m DEB = 45 and m AED = m CEB = 135. Module 4 165 Lesson 1
Explain 1 Using Verical Angles You can use he Verical Angles Theorem o find missing angle measures in siuaions involving inersecing lines. Example 1 Cross braces help keep he deck poss sraigh. Find he measure of each angle. 5 146 6 7 6 Because verical angles are congruen, m 6 = 146. 5 and 7 From Par A, m 6 = 146. Because 5 and 6 form a linear pair, hey are supplemenary and m 5 = 180-146 = 34. m 7 = 34 because 7 also forms a linear pair wih 6, or because i is a verical angle wih 5. Your Turn 6. The measures of wo verical angles are 58 and (3x + 4). Find he value of x. 58 = 3x + 4 54 = 3x 18 = x 7. The measures of wo verical angles are given by he expressions (x + 3) and (2x - 7). Find he value of x. Wha is he measure of each angle? x + 3 = 2x - 7 x + 10 = 2x 10 = x The measure of each angle is (x + 3) = (10 + 3) = 13. Module 4 166 Lesson 1
Explain 2 Using Supplemenary and Complemenary Angles Recall wha you know abou complemenary and supplemenary angles. Complemenary angles are wo angles whose measures have a sum of 90. Supplemenary angles are wo angles whose measures have a sum of 180. You have seen ha wo angles ha form a linear pair are supplemenary. Example 2 Use he diagram below o find he missing angle measures. Explain your reasoning. B A C 50 F D E Find he measures of AFC and AFB. AFC and CFD are a linear pair formed by an inersecing line and ray, AD and FC, so hey are supplemenary and he sum of heir measures is 180. By he diagram, m CFD = 90, so m AFC = 180-90 = 90 and AFC is also a righ angle. Because ogeher hey form he righ angle AFC, AFB and BFC are complemenary and he sum of heir measures is 90. So, m AFB = 90 - m BFC = 90-50 = 40. Find he measures of DFE and AFE. BFA and DFE are formed by wo inersecing lines and are opposie each oher, so he angles are verical angles. So, he angles are congruen. From Par A m AFB = 40, so m DFE = 40 also. Because BFA and AFE form a linear pair, he angles are supplemenary and he sum of heir measures is 180. So, m AFE = 180 - m BFA = 180-40 = 140. 8. In Par A, wha do you noice abou righ angles AFC and CFD? Make a conjecure abou righ angles. Possible answer: Boh angles have measure 90. Conjecure: All righ angles are congruen. Module 4 167 Lesson 1
Your Turn You can represen he measures of an angle and is complemen as x and (90 - x). Similarly, you can represen he measures of an angle and is supplemen as x and (180 - x). Use hese expressions o find he measures of he angles described. 9. The measure of an angle is equal o he measure of is complemen. 10. The measure of an angle is wice he measure of is supplemen. x = 90 - x 2x = 90 x = 45; so, 90 - x = 45 The measure of he angle is 45 he measure of is complemen is 45. x = 2 (180 - x) x = 360-2x 3x = 360 x = 120; so, 180 - x = 60 The measure of he angle is 120 he measure of is supplemen is 60.
Name Class Dae 4.2 Transversals and Parallel Lines Essenial Quesion: How can you prove and use heorems abou angles formed by ransversals ha inersec parallel lines? Explore Exploring Parallel Lines and Transversals A ransversal is a line ha inersecs wo coplanar lines a wo differen poins. In he figure, line is a ransversal. The able summarizes he names of angle pairs formed by a ransversal. Resource Locker 1 2 4 3 5 6 8 7 p q Angle Pair Example Image Credis: Ruud Morijn Phoographer/Shuersock Corresponding 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 opposie sides of he ransversal beween he inerseced lines. Alernae exerior angles lie on opposie sides of he ransversal and ouside he inerseced lines. Recall ha parallel lines lie in he same plane and never inersec. In he figure, line l is parallel o line m, wrien lǁm. The arrows on he lines also indicae ha hey are parallel. l m l m 1 and 5 3 and 6 3 and 5 1 and 7 Module 4 175 Lesson 2 DO NOT EDIT--Changes mus be made hrough File info CorrecionKey=NL-C;CA-C ae
When parallel lines are cu by a ransversal, he angle pairs formed are eiher congruen or supplemenary. The following posulae is he saring poin for proving heorems abou parallel lines ha are inerseced by a ransversal. Same-Side Inerior Angles Posulae If wo parallel lines are cu by a ransversal, hen he pairs of same-side inerior angles are supplemenary. Follow he seps o illusrae he posulae and use i o find angle measures. A B C D Draw wo parallel lines and a ransversal, and number he angles formed from 1 o 8. 1 2 p 4 3 5 6 q 8 7 Idenify he pairs of same-side inerior angles. 4 and 5; 3 and 6 Wha does he posulae ell you abou hese same-side inerior angle pairs? Given p q, hen 4 and 5 are supplemenary and 3 and 6 are supplemenary. If m 4 = 70, wha is m 5? Explain. m 5 = 110 ; 4 and 5 are supplemenary, so m 4 + m 5 = 180. Therefore 70 + m 5 = 180, so m 5 = 110. 1. Explain how you can find m 3 in he diagram if p q and m 6 = 61. 3 and 6 are supplemenary, so m 3 + m 6 = 180. Therefore m 3 + 61 = 180, so m 3 = 119. 2. Wha If? If m n, how many pairs of same-side inerior angles are shown in he figure? Wha are he pairs? Two pairs; 3 and 5, 4 and 6 m 1 3 2 4 5 6 8 n 7 Module 4 176 Lesson 2
Explain 1 Proving ha Alernae Inerior Angles are Congruen Oher pairs of angles formed by parallel lines cu by a ransversal are alernae inerior angles. Alernae Inerior Angles Theorem If wo parallel lines are cu by a ransversal, hen he pairs of alernae inerior angles have he same measure. To prove somehing o be rue, you use definiions, properies, posulaes, and heorems ha you already know. Example 1 Prove he Alernae Inerior Angles Theorem. Given: p q Prove: m 3 = m 5 1 2 4 3 5 6 8 7 q p Complee he proof by wriing he missing reasons. Choose from he following reasons. You may use a reason more han once. Same-Side Inerior Angles Posulae Subracion Propery of Equaliy Given Definiion of supplemenary angles Subsiuion Propery of Equaliy Linear Pair Theorem Saemens 1. p q 2. 3 and 6 are supplemenary. 3. m 3 + m 6 = 180 Reasons 1. Given 2. Same-Side Inerior Angles Posulae 3. Definiion of supplemenary angles 4. 5 and 6 are a linear pair. 5. 5 and 6 are supplemenary. 6. m 5 + m 6 = 180 7. m 3 + m 6 = m 5 + m 6 8. m 3 = m 5 4. Given 5. Linear Pair Theorem 6. Definiion of supplemenary angles 7. Subsiuion Propery of Equaliy 8. Subracion Propery of Equaliy 3. In he figure, explain 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 Propery of Equaliy). Module 4 177 Lesson 2
4. Suppose 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 supplemenary, so m 5 = 123. Since 5 and 6 are supplemenary, m 6 = 57. Explain 2 Proving ha Corresponding Angles are Congruen Two parallel lines cu by a ransversal also form angle pairs called corresponding angles. Corresponding Angles Theorem If wo parallel lines are cu by a ransversal, hen he pairs of corresponding angles have he same measure. Example 2 Complee a proof in paragraph form for he Corresponding Angles Theorem. Given: p q Prove: m 4 = m 8 1 2 4 3 5 6 8 7 q p By he given saemen, p q. 4 and 6 form a pair of alernae inerior angles. So, using he Alernae Inerior Angles Theorem, m 4 = m 6. 6 and 8 form a pair of verical angles. So, using he Verical Angles Theorem, m 6 = m 8. Using he Subsiuion Propery of Equaliy in m 4 = m 6, subsiue m 4 for m 6. The resul is m 4 = m 8. 5. Use he diagram in Example 2 o explain how you can prove he Corresponding Angles Theorem using he Same-Side Inerior Angles Posulae and a linear pair of angles. By he Same-Side Inerior Angles Theorem, m 4 + m 5 = 180. As a linear pair, m 4 + m 1 = 180. Therefore m 4 + m 1 = m 4 + m 5, so m 1 = m 5. 6. Suppose m 4 = 36. Find m 5. Explain. m 5 = 144 ; Since 1 and 4 form a linear pair, hey are supplemenary. So, m 1 = 144. Using he Corresponding Angles Theorem, you know ha m 1 = m 5. So, m 5 = 144. Module 4 178 Lesson 2