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

Holt Geometry 3-3 Proving Lines Parallel 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.

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

Holt Geometry 3-3 Proving Lines Parallel 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.

Holt Geometry 3-3 Proving Lines Parallel

Holt Geometry 3-3 Proving Lines Parallel Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m. #1 1  5 1  5 1 and 5 are corresponding angles. ℓ || m Conv. of Corr. s Post.

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

Holt Geometry 3-3 Proving Lines Parallel # 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 || m Conv. of Corr. s Post.

Holt Geometry 3-3 Proving Lines Parallel # 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 ℓ || m Conv. of Corr. s Post. 7  5 m7 = m5

Holt Geometry 3-3 Proving Lines Parallel

Holt Geometry 3-3 Proving Lines Parallel Use the given information and the theorems you have learned to show that r || s. # 1 2  62  6 2  62 and 6 are alternate interior angles. r || sConv. Of Alt. Int. s Thm.

Holt Geometry 3-3 Proving Lines Parallel m6 = (6x + 18)°, m7 = (9x + 12)°, x = 10 Use the given information and the theorems you have learned to show that r || s. # 2 m6 = 6x + 18 = 6(10) + 18 = 78 m7 = 9x + 12 = 9(10) + 12 = 102

Holt Geometry 3-3 Proving Lines Parallel Use the given information and the theorems you have learned to show that r || s. #2 (Continued) r || s Conv. of Same-Side Int. s Thm. m6 + m7 = 78° + 102° = 180°6 and 7 are same-side interior angles. m6 = (6x + 18)°, m7 = (9x + 12)°, x = 10

Holt Geometry 3-3 Proving Lines Parallel # 3 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. Ext. s Thm. 4  8 Congruent angles

Holt Geometry 3-3 Proving Lines Parallel # 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 = 100 and m7 = 100 3  7r||s Conv. of the Alt. Int. s Thm. m3 = 2x = 2(50) = 100° m7 = x + 50 = = 100°

Holt Geometry 3-3 Proving Lines Parallel Proof # 1 Given: l || m, 1  3 Prove: p || r

Holt Geometry 3-3 Proving Lines Parallel Proof 1 Continued StatementsReasons 1. l || m 5. r ||p 2. 1  3 3. 1  2 4. 2  3 2. Given 1. Given 3. Corresponding < Post. 4. Transitive POC 5. Conv. of Alt. Ext s Thm.

Holt Geometry 3-3 Proving Lines Parallel Proof # 2 Given: 1  4, 3 and 4 are supplementary. Prove: ℓ || m

Holt Geometry 3-3 Proving Lines Parallel Proof 2 Continued StatementsReasons 7. m<2 = m<3 1. 1  4 1. Given 2. < 3 and < 4 are supp2. 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  36. Vert.s Thm. 8. m2 + m1 = 180 8. Substitution 9. ℓ || m 9. Conv. of Same-Side Interior s Thm. 7. Def of  <s

Holt Geometry 3-3 Proving Lines Parallel Proof # 3 Given: 1 and 3 are supplementary. Prove: m || n

Holt Geometry 3-3 Proving Lines Parallel Proof 3 Continued StatementsReasons 1. <1 and <3 are supp 2. 2 and 3 are supp 3. 1  2 4. m ||n 2. Linear Pair Thm 1. Given 3.  Supplements Thm. 4. Conv. Of Corr < Post.

Holt Geometry 3-3 Proving Lines Parallel 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.

Holt Geometry 3-3 Proving Lines Parallel 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.

Holt Geometry 3-3 Proving Lines Parallel Exit Question Complete Exit Questions to be turned in –This is not graded

Holt Geometry 3-3 Proving Lines Parallel Homework Page : # 1-10 & # 22 Chapter 3 Test on Friday December 21st (C-level)