Holt McDougal Geometry 4-7-ext Lines and Slopes 4-7-ext Lines and Slopes Holt Geometry Lesson...
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Transcript of Holt McDougal Geometry 4-7-ext Lines and Slopes 4-7-ext Lines and Slopes Holt Geometry Lesson...
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