Proving Lines Parallel Chapter 3-5 Proving Lines Parallel

Lesson 3-5 Ideas/Vocabulary Recognize angle conditions that occur with parallel lines. Prove that two lines are parallel based on given angle relationships. Lesson 3-5 Ideas/Vocabulary

Transitive property of Parallels If two lines are parallel to the same line, then they are parallel to each other. If p // q and q // r, then p // r. p q r

Reminders from Section 1 We will use these same theorems to prove the lines are parallel given certain angle information.

Corresponding Angle Theorem If two parallel lines are cut by a transversal, then corresponding angles are congruent. // lines  corresponding s are 

Corresponding Angle Theorem

Alternate Interior Angle Theorem If two parallel lines are cut by a transversal, then alternate interior angles are congruent. // lines  Alt. Int. s are 

Alternate Interior Angle Theorem

Alternate Exterior Angle Theorem If two parallel lines are cut by a transversal, then alternate exterior angles are congruent. // lines  Alt. Ext. s are 

Alternate Exterior Angle Theorem

Consecutive Interior Angle Theorem If two parallel lines are cut by a transversal, then consecutive interior angles are supplementary. // lines  Consec. Int. s are Supp.

Consecutive Interior Angle Theorem 1 2 m1 + m2 = 180

Two  Theorem If two lines are perpendicular to the same line, then they are parallel to each other. If m  p and n  p, then m // n. p m n

Animation: Construct a Parallel Line Through a Point not on Line Lesson 3-5 Postulates

Lesson 3-5 Theorems

 a//b  a is not // c  b is not // c 77o Identify Parallel Lines Determine which lines, if any, are parallel. Consec. Int. s are supp. 77o  a//b Alt. Int. s are not   a is not // c Consec. Int. s are not supp.  b is not // c Lesson 3-5 Example 1

Determine which lines, if any are parallel. I. e || f II. e || g III Determine which lines, if any are parallel. I. e || f II. e || g III. f || g A B C D I only II only III only I, II, and III Lesson 3-5 CYP 1

Solve Problems with Parallel Lines ALGEBRA Find x and m ZYN so that || . Explore From the figure, you know that m WXP = 11x – 25 and m ZYN = 7x + 35. You also know that WXP and ZYN are alternate exterior angles. Lesson 3-5 Example 2

ALGEBRA Find x and m ZYN so that || . If Alt. Ext. angles are , then the lines will be // m WXP = m ZYN Alternate exterior  thm. 11x – 25 = 7x + 35 Substitution 4x – 25 = 35 Subtract 7x from each side. 4x = 60 Add 25 to each side. x = 15 Divide each side by 4. Lesson 3-5 Example 2

Solve Problems with Parallel Lines Now use the value of x to find m ZYN. m ZYN = 7x + 35 Original equation = 7(15) + 35 x = 15 = 140 Simplify. Answer: x = 15, m ZYN = 140 Lesson 3-5 Example 2

ALGEBRA Find x so that || . C D x = 60 x = 9 x = 12 Lesson 3-5 CYP 2

Prove Lines Parallel Prove: r || s Given: ℓ || m Lesson 3-5 Example 3

Prove Lines Parallel Proof: Statements Reasons 1. 1. Given 2. 2. Consecutive Interior Angle Theorem 3. 3. Definition of supplementary angles 4. 4. Definition of congruent angles 5. 5. Substitution 6. 6. Definition of supplementary angles 7. 7. If consecutive interior angles theorem Lesson 3-5 Example 3

not enough information to determine Given x || y and , can you use the Corresponding Angles Postulate to prove a || b? A B C yes no not enough information to determine Lesson 3-5 CYP 3

Slope and Parallel Lines Determine whether p || q. slope of p: slope of q: Answer: Since the slopes are equal, p || q. Lesson 3-5 Example 4

Determine whether r || s. A B C Yes, r is parallel to s. No, r is not parallel to s. It cannot be determined. Lesson 3-5 CYP 4