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

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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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