Proving Lines Parallel 3.4

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

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.

Given: 4  8 Prove: ℓ || m. 4  8 4  8 4 and 8 are corresponding angles. ℓ || m Converse of Corresponding s Postulate

Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m. m3 = (4x – 80)°, m7 = (3x – 50)°, x = 30 m3 = 4(30) – 80 = 40Substitute 30 for x. m7 = 3(30) – 50 = 40Substitute 30 for x. ℓ || m Converse of Corresponding s Postulate 3  7 Definition of Congruence m3 = m7 Substitution Prop. of Equality

Use 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.

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 = 77 Substitute 13 for x. ℓ || m Conv. of Corr. s Post. 7  5 Def. of  s. m7 = m5 Subs. Prop. of Eq.

Use the given information and the theorems you have learned to show that r || s. 4  8 4  84 and 8 are alternate exterior angles. r || sConv. of Alt. Ext. s Thm.

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. m2 = 10x + 8 = 10(5) + 8 = 58Substitute 5 for x. m3 = 25x – 3 = 25(5) – 3 = 122Substitute 5 for x. r || s Conv. of Same-Side Int. s Thm. m2 + m3 = 58° + 122° = 180°2 and 3 are same-side interior angles.

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 = = 100° Substitute 50 for x.

StatementsReasons 1. p || r 5. ℓ ||m 2. 3  2 3. 1  3 4. 1  2 2. Alternate Exterior s Thm. 1. Given 3. Given 4. Substitution Property 5. Conv. of Corr. s Post. Given: p || r, 1  3 Prove: ℓ || m

Given: 1  4 & 3 and 4 are supplementary. Prove: ℓ || m StatementsReasons 1. 1  4 1. Given 2. m1 = m4 2. Definition of congruence 3. 3 and 4 are supp. 3. Given 4. m3 + m4 = 180 4. Def. of Supplementary 5. m3 + m1 = 180 5. Substitution 6. m2 = m36. Vertical s Theorem 7. m2 + m1 = 180 7. Substitution 8. ℓ || m 8. Conv. of Cons. Int. s

StatementsReasons 1. m || n 1. Given 2. 1  2 3. n || k 4. 2  34. Corresponding ’s Post. 5. 1  3 5. Transitive Property 6. m || k6. Converse of Corresponding ’s Post. Given: m||n and n||k Prove: m||k m n 1 k Corresponding ’s Post. 3. Given

StatementsReasons 1. m1 + m3 =180 1. Given 2. m2  m3 4. m1 + m2 = 180 5. m || n 5. Converse Consecutive Interior Angles Theorem 2. Vertical Angles Theorem 4. Substitution Property Given: m  1 + m  3 = 180 . Prove: Lines m and n are parallel. 3. m2 = m3 3. Def. of Congruence

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. Substitute 15 for x in each expression. m1 = 8x + 20 = 8(15) + 20 = 140 m2 = 2x + 10 = 2(15) + 10 = 40 m1+m2 = 180 1 and 2 are supplementary. The consecutive interior angles are supplementary, so pieces A and B are parallel by the Converse of the Consecutive Interior Angles Theorem.

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° The angles are congruent, so the oars are || by the Converse of the Corresponding s Post. 3y + 6 = 3(8) + 6 = 30°

Name the postulate or theorem that proves p || r. 1. 4   5Converse of Alternate Interior  s Theorem 2.  2   7 Converse of Alternate Exterior s Theorem 3.  3   7Converse of Corresponding  s Postulate 4.  3 and  5 are supplementary. Converse of Consecutive Interior  s Theorem

Use the theorems and given information to prove p || r. m2 = (5x + 20)°, m  7 = (7x + 8)°, and x = 6 m2 = 5(6) + 20 = 50° m7 = 7(6) + 8 = 50° m2 = m7, so 2 ≅ 7 p || r by the Converse of Alternate Exterior s Theorem

Assignment 3-4 Proof with Parallel Lines