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3-3

In this lesson, we explore how to use the angles formed by a transversal to prove that two lines are parallel. We'll state the converse of various geometric statements and understand the importance of the converse. By applying the Converse of the Corresponding Angles Postulate and other angle relationships, students will learn to demonstrate the parallelism of lines using given angle measures. The lesson includes practice problems, examples, and a quiz to ensure understanding.

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3-3

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  1. 3-3 Proving Lines Parallel Warm Up Lesson Presentation Lesson Quiz Holt Geometry

  2. Warm Up State the converse of each statement. 1.If a = b, then a + c = b + c. 2. If mA + mB = 90°, then A and B are complementary. 3. If AB + BC = AC, then A, B, and C are collinear. If a + c = b + c, then a = b. If A and B are complementary, then mA + mB =90°. If A, B, and C are collinear, then AB + BC = AC.

  3. Objective Use the angles formed by a transversal to prove two lines are parallel.

  4. Recall that the converse of a theorem is found by exchanging the hypothesis and conclusion. The converse of a theorem is not automatically true. If it is true, it must be stated as a postulate or proved as a separate theorem.

  5. #1 Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m. 1 5 1 5 1 and 5 are corresponding angles. ℓ || mConv. of Corr. s Post.

  6. # 2 Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m. m4 = (2x + 10)°, m8 = (3x – 55)°, x = 65 m4 = 2(65) + 10 = 140 m8 = 3(65) – 55 = 140 m4 = m8 4  8 ℓ || m Conv. of Corr. s Post.

  7. # 3 Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m. m1 = m3 1 3 1 and 3 are corresponding angles. ℓ || mConv. of Corr. s Post.

  8. # 4 Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m. m7 = (4x + 25)°, m5 = (5x + 12)°, x = 13 m7 = 4(13) + 25 = 77 m5 = 5(13) + 12 = 77 m7 = m5 7  5 ℓ || m Conv. of Corr. s Post.

  9. # 1 Use the given information and the theorems you have learned to show that r || s. 26 2 6 2 and 6 are alternate interior angles. r || sConv. Of Alt. Int. s Thm.

  10. # 2 Use the given information and the theorems you have learned to show that r || s. m6 = (6x + 18)°, m7 = (9x + 12)°, x = 10 m6 = 6x + 18 = 6(10) + 18 = 78 m7 = 9x + 12 = 9(10) + 12 = 102

  11. #2 (Continued) Use the given information and the theorems you have learned to show that r || s. m6 = (6x + 18)°, m7 = (9x + 12)°, x = 10 m6 + m7 = 78° + 102° = 180°6 and 7 are same-side interior angles. r || sConv. of Same-Side Int. s Thm.

  12. # 3 Refer to the diagram. Use the given information and the theorems you have learned to show that r || s. m4 = m8 4 8 Congruent angles 4 8 4 and 8 are alternate exterior angles. r || sConv. of Alt. Ext. s Thm.

  13. # 4 Refer to the diagram. Use the given information and the theorems you have learned to show that r || s. m3 = 2x, m7 = (x + 50), x = 50 m3 = 2x = 2(50) = 100° m7 = x + 50 = 50 + 50 = 100° m3 =100 and m7 =100 3  7 r||sConv. of the Alt. Int. s Thm.

  14. Proof # 1 Given:l || m , 1 3 Prove: p || r

  15. Proof 1 Continued 1. Given 1.l || m 2.1  3 2. Given 3.1  2 3. Corresponding < Post. 4.2  3 4. Transitive POC 5. r ||p 5. Conv. of Alt. Ext s Thm.

  16. Proof # 2 Given: 1 4, 3 and 4 are supplementary. Prove: ℓ || m

  17. Proof 2 Continued 1. Given 1.1  4 2. < 3 and < 4 are supp 2.Given 3.m<1 = m<4 3.Def of  <s 4. m3 + m4 = 180 4. Def. of Supp <s 5. m3 + m1 = 180 5. Substitution 6.2  3 6. Vert.s Thm. 7. Def of  <s 8. m2 + m1 = 180 8. Substitution 9. Conv. of Same-Side Interior sThm. 9. ℓ || m

  18. Proof # 3 Given: 1 and 3 are supplementary. Prove: m || n

  19. Proof 3 Continued 1. Given 1.<1 and <3 are supp 2.2 and 3 are supp 2. Linear Pair Thm 3.1  2 3. Supplements Thm. 4. m ||n 4. Conv. Of Corr < Post.

  20. Ms. Anderson’s Rowing Example # 1 During a race, all members of a rowing team should keep the oars parallel on each side. If on the port side m < 1 = (3x + 13)°, m<2 = (5x - 5)°, & x= 9. Show that the oars are parallel. 3x+13 = 3(9) + 13 = 40° 5x - 5 = 5(9) - 5 = 40° The angles are congruent, so the oars are || by the Conv. of the Corr. s Post.

  21. Ms. Anderson’s Rowing Example # 2 Suppose the corresponding angles on the starboard side of the boat measure (4y – 2)° and (3y + 6)°, & y = 8. Show that the oars are parallel. 4y – 2 = 4(8) – 2 = 30° 3y + 6 = 3(8) + 6 = 30° The angles are congruent, so the oars are || by the Conv. of the Corr. s Post.

  22. Exit Question • Complete Exit Questions to be turned in • This is not graded

  23. Homework • Page 166-167: # 1-10 & # 22 • Chapter 3 Test on Friday December 21st (C-level)

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