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. Standard 7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles. (Key) Standard 16.0 Students perform basic constructions with a straightedge and compass, such as angle bisectors, perpendicular bisectors, and the line parallel to a given line through a point off the line. (Key)

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

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

Lesson 3-5 Theorems

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

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

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

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

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

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

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

Lesson 3-5 Example 3 Prove Lines Parallel 2.2. Consecutive Interior Angle Theorem Substitution 6.6. Definition of supplementary angles 7.7. If consecutive interior angles theorem 1.1. Given Proof: StatementsReasons Definition of congruent angles 3.3. Definition of supplementary angles

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

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

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

Homework Chapter 3-5 Pg – 5, 7 – 19, 23 (proof), 24 (proof), 37, 50 – 52