Topic: Proving Lines Parallel or Perpendicular

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Please please please answer it completely. Please Make sure your handwriting is legible. Please please please make it neat and understandable. Inx Proving Lines Parallel or Perpendicular LESSON 8.2 PROVING LINES PARALLEL OR PERPENDICULAR What You'll Learn Connect To determine and prove the conditions under which lines Now, you will learn if the and segments are parallel or perpendicular. converse of each of the theorems discussed in Lesson 8.1 is true. Explore BAY 1st Avenue 120 Plan for a New Village 2nd Avenue 120 The new exclusive village will consist of 60 houses on three parallel streets. The sewer line is marked to show that it does not lead to the bay. How do you know based 3rd Avenue on this plan that the streets are parallel? What kind of angles are the two 60 angles marked on the diagram? What do you know about these angles? What kind of angles are the two 120" angles marked on the diagram? What do you know about these angles? Now, let us discuss the theorems that prove that two lines are parallel. These theorems are converses of the theorems presented in Lesson 8.1. POSTULATE If two lines are cut by a transversal and a pair of corresponding angles are congruent, then the lines are parallel. LESSON 8.2 Proving Lines Parallel or Perpendicular 393We will use the postulate above to prove the next theorem. THEOREM 8-4 If two lines are cut by a transversal and a pair of alternate interior angles are congruent, then the lines are parallel. Given: 41 = 42 Prove : rlIs Proof : Statements Reasons 1. 41= 42 1. Given 2. 42 = 23 2. Vertical Zs are =. 3. 41= 23 3. Congruence of Zs is transitive. 4. If two lines are cut by a transversal and a pair of 4. rlls corresponding angles are =, then the lines are II. The converses of Alternate Exterior Angles Theorem and Same-side Interior ? Angle Theorem are stated. Think About This THEOREM 8-5 If two lines If two lines are cut by a transversal and a pair of alternate exterior angles are cut by two are congruent, then the lines are parallel. transversals, what is the smallest number of angles that you need THEOREM 8-6 to measure to guarantee that the If two lines are cut by a transversal and a pair of same-side interior angles lines are parallel? are supplementary, then the lines are parallel. The proof of these theorems are given in Exercises 26 and 27. 394 CHAPTER 8 Parallelism and PerpendicularityEXAMPLE Use the given information to determine which lines are parallel. a. 41= 43 b. m24 + m25 = m46 c. m42= m/4 + m/5 SOLUTION: a. rlls If two lines are cut by a transversal and a pair of corresponding Ls are =, then the lines are II. b. pllq If two lines are cut by a transversal and a pair of alternate interior Ls are #, then the lines are II. c. rlIs If two lines are cut by a transversal and a pair of alternate interior Zs are =, then the lines are II. Try This Use the given information to determine which lines are parallel. a. 41= 42 b. 41 = 23 c. 43 = 24 Here are two other theorems involving parallel lines. THEOREM 8-7 In a plane, two coplanar lines perpendicular to the same line are parallel. THEOREM 8-8 Two lines parallel to a third line are parallel to each other. The proof of these theorems are given in Exercises 28 and 29. LESSON 8.2 Proving Lines Parallel or Perpendicular 395Ways to Prove Two Lines are Parallel Show that... 1 . a pair of corresponding angles are congruent. 2 . a pair of alternate interior angles are congruent. 3 . a pair of alternate exterior angles are congruent. 4. a pair of same-side interior angles are supplementary. . two coplanar lines are perpendicular to the same line. MorTUTOR ibrtogas Reflect Have you always wanted to try something but was afraid to do so such as ice skating, biking, or diving? The first attempts will likely be unsuccessful and may be embarrassing. But how will you learn if you do not try? Exercises Comprehension Check 18 MEROENT Determine which pair of lines, if any, are parallel. Explain. 1 3. 110 - m 85 85 60 - n 8-8 MERON 2. 4. 570 p pabroad at novig s 123 396 CHAPTER 8 Parallelism and Perpendicularity5 . 98* 6. it C funnotel moving andt outs 75 75. isues a postulate. Ilse the given information to decide which lines are parallel. Justify your answers with a theorem or 7. 25 = 410 8. 42 = 24 2 P 9. m210 + m