Holt McDougal Geometry

4-7-ext Lines and Slopes4-7-ext Lines and Slopes

Holt Geometry

Lesson PresentationLesson Presentation

Holt McDougal Geometry

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 L1 || L2 and L1 and L2 are not vertical, then m1 = m2.

2. If m1 = m2, then L1 || L2.

3. If L1 and L2 are vertical, then L1 || L2.

Holt McDougal Geometry

4-7-ext Lines and Slopes

Example 1:Proving the Parallel Lines Theorem

Are the lines parallel? Explain.

No ; 43

= 54

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: m1 = m2

Prove: L1 || L2

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 ; 32

= 34

- = 98

-1

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 m1 = Then m2 = - . Draw PQR with

sides of length a and b and a right angle at R to

represent the rise and run of L1, and PST with

sides of length b and a to represent the rise and run

of L2

ab

ba

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.

L1 ⊥ L2 by h.

Substitution Property of Equality

Angle Addition Postulate

Def. of Perpendicular Lines