Holt McDougal Geometry 4-7-ext Lines and Slopes 4-7-ext Lines and Slopes Holt Geometry Lesson Presentation Lesson Presentation Holt McDougal Geometry

4-7-ext Lines and Slopes Prove the slope criteria for parallel and perpendicular lines. Objective

Holt McDougal Geometry 4-7-ext Lines and Slopes Slopes can be used to determine if two lines in a coordinate plane are parallel or perpendicular. In this lesson, you will prove the Parallel Lines Theorem and the Perpendicular Lines Theorem. Suppose that L1 and L2 are two lines in the coordinate plane with slopes m1 and m2. The proof of the Parallel Lines Theorem can be broken into three parts: 1. If L 1 || L 2 and L 1 and L 2 are not vertical, then m 1 = m If m 1 = m 2, then L 1 || L If L 1 and L 2 are vertical, then L 1 || L 2.

Holt McDougal Geometry 4-7-ext Lines and Slopes Example 1:Proving the Parallel Lines Theorem Are the lines parallel? Explain. No ; 4 3 = 5 4

Holt McDougal Geometry 4-7-ext Lines and Slopes Check It Out! Example 1 Complete the two-column proof, using the figure in Example 1 Given: m 1 = m 2 Prove: L 1 || L 2 Proof:

Holt McDougal Geometry 4-7-ext Lines and Slopes Check It Out! Example 1 continue Given Definition of slope Substitution Property of Equality Multiplication Property of Equality SAS CPCTC Converse of Corr. Angles Postulate

Holt McDougal Geometry 4-7-ext Lines and Slopes Example 2 : Proving the Perpendicular Lines Are the lines perpendicular? Explain. No ; 3 2 = = 9 8

Holt McDougal Geometry 4-7-ext Lines and Slopes Check It Out! Example 2 Complete the paragraph proof below. Given: m1 · m2 = -1 Prove: L1 ⊥ L2 Proof: Let m 1 = Then m 2 = -. Draw PQR with sides of length a and b and a right angle at R to represent the rise and run of L 1, and PST with sides of length b and a to represent the rise and run of L 2 a b b a

Holt McDougal Geometry 4-7-ext Lines and Slopes Check It Out! Example 2 Continued PQR PST by a., so ∠ QPR ∠ SPT because b., m ∠ QPR = m ∠ SPT by c. By construction, PT ⊥ PR, so m ∠ RPT = 90° by the definition of perpendicular lines. m ∠ QPT + m ∠ QPR = 90° by e. = ~ = ~ SAS CPCTC Def. of Congruent Angle Def. of Complementrary angle or Angle Addition Postulate

Holt McDougal Geometry 4-7-ext Lines and Slopes Check It Out! Example 2 Continued Then m ∠ QPT + m ∠ SPT = 90° by f. so m ∠ SPQ = 90° by g. L 1 ⊥ L 2 by h. Substitution Property of Equality Angle Addition Postulate Def. of Perpendicular Lines