1. Holt Geometry 3-3 Proving Lines Parallel Warm Up Identify each angle pair. 1. 1 and 3 2. 3 and 6 3. 4 and 5 4. 6 and 7 same-side int s corr. s alt. int. s alt. ext. s

2. Holt Geometry 3-3 Proving Lines Parallel Use the angles formed by a transversal to prove two lines are parallel. Objective

3. Holt Geometry 3-3 Proving Lines Parallel

4. Holt Geometry 3-3 Proving Lines Parallel Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m. Example 1A: Using the Converse of the Corresponding Angles Postulate 4  8 4  8 4 and 8 are corresponding angles. ℓ || m Conv. of Corr. s Post.

5. Holt Geometry 3-3 Proving Lines Parallel Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m. Example 1B: Using the Converse of the Corresponding Angles Postulate m3 = (4x – 80)°, m7 = (3x – 50)°, x = 30 m3 = 4(30) – 80 = 40Substitute 30 for x. m8 = 3(30) – 50 = 40Substitute 30 for x. ℓ || m Conv. of Corr. s Post. 3  8 Def. of  s. m3 = m8Trans. Prop. of Equality

6. Holt Geometry 3-3 Proving Lines Parallel Check It Out! Example 1a 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 || m Conv. of Corr. s Post.

7. Holt Geometry 3-3 Proving Lines Parallel Check It Out! Example 1b 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 = 77Substitute 13 for x. m5 = 5(13) + 12 = 77Substitute 13 for x. ℓ || m Conv. of Corr. s Post. 7  5 Def. of  s. m7 = m5Trans. Prop. of Equality

8. Holt Geometry 3-3 Proving Lines Parallel The Converse of the Corresponding Angles Postulate is used to construct parallel lines. The Parallel Postulate guarantees that for any line ℓ, you can always construct a parallel line through a point that is not on ℓ.

9. Holt Geometry 3-3 Proving Lines Parallel

10. Holt Geometry 3-3 Proving Lines Parallel Use the given information and the theorems you have learned to show that r || s. Example 2A: Determining Whether Lines are Parallel 4  8 4  84 and 8 are alternate exterior angles. r || sConv. Of Alt. Int. s Thm.

11. Holt Geometry 3-3 Proving Lines Parallel m2 = (10x + 8)°, m3 = (25x – 3)°, x = 5 Use the given information and the theorems you have learned to show that r || s. Example 2B: Determining Whether Lines are Parallel m2 = 10x + 8 = 10(5) + 8 = 58Substitute 5 for x. m3 = 25x – 3 = 25(5) – 3 = 122Substitute 5 for x.

12. Holt Geometry 3-3 Proving Lines Parallel m2 = (10x + 8)°, m3 = (25x – 3)°, x = 5 Use the given information and the theorems you have learned to show that r || s. Example 2B Continued r || s Conv. of Same-Side Int. s Thm. m2 + m3 = 58° + 122° = 180°2 and 3 are same-side interior angles.

13. Holt Geometry 3-3 Proving Lines Parallel Check It Out! Example 2a m4 = m8 Refer to the diagram. Use the given information and the theorems you have learned to show that r || s. 4  84 and 8 are alternate exterior angles. r || sConv. of Alt. Int. s Thm. 4  8 Congruent angles

14. Holt Geometry 3-3 Proving Lines Parallel Check It Out! Example 2b 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 = 100 and m7 = 100 3  7r||s Conv. of the Alt. Int. s Thm. m3 = 2x = 2(50) = 100°Substitute 50 for x. m7 = x + 50 = 50 + 50 = 100° Substitute 5 for x.

15. Holt Geometry 3-3 Proving Lines Parallel Example 3: Proving Lines Parallel Given: p || r, 1  3 Prove: ℓ || m

16. Holt Geometry 3-3 Proving Lines Parallel Example 3 Continued StatementsReasons 1. p || r 5. ℓ ||m 2. 3  2 3. 1  3 4. 1  2 2. Alt. Ext. s Thm. 1. Given 3. Given 4. Trans. Prop. of  5. Conv. of Corr. s Post.

17. Holt Geometry 3-3 Proving Lines Parallel Check It Out! Example 3 Given: 1  4, 3 and 4 are supplementary. Prove: ℓ || m

18. Holt Geometry 3-3 Proving Lines Parallel Check It Out! Example 3 Continued StatementsReasons 1. 1  4 1. Given 2. m1 = m42. Def.  s 3. 3 and 4 are supp. 3. Given 4. m3 + m4 = 1804. Trans. Prop. of  5. m3 + m1 = 180 5. Substitution 6. m2 = m36. Vert.s Thm. 7. m2 + m1 = 180 7. Substitution 8. ℓ || m8. Conv. of Same-Side Interior s Post.

19. Holt Geometry 3-3 Proving Lines Parallel Example 4: Carpentry Application A carpenter is creating a woodwork pattern and wants two long pieces to be parallel. m1= (8x + 20)° and m2 = (2x + 10)°. If x = 15, show that pieces A and B are parallel.

20. Holt Geometry 3-3 Proving Lines Parallel Example 4 Continued A line through the center of the horizontal piece forms a transversal to pieces A and B. 1 and 2 are same-side interior angles. If 1 and 2 are supplementary, then pieces A and B are parallel. Substitute 15 for x in each expression.

21. Holt Geometry 3-3 Proving Lines Parallel Example 4 Continued m1 = 8x + 20 = 8(15) + 20 = 140 m2 = 2x + 10 = 2(15) + 10 = 40 m1+m2 = 140 + 40 = 180 Substitute 15 for x. 1 and 2 are supplementary. The same-side interior angles are supplementary, so pieces A and B are parallel by the Converse of the Same-Side Interior Angles Theorem.

22. Holt Geometry 3-3 Proving Lines Parallel Check It Out! Example 4 What if…? Suppose the corresponding angles on the opposite side of the boat measure (4y – 2)° and (3y + 6)°, where 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.

23. Holt Geometry 3-3 Proving Lines Parallel Homework: p.166-169, #12-22 Even, 24-31, 34, 35, 38, 40, 41, 55, 56, 60