Proving Lines Parallel

1
Chapter 3-5

• Proving Lines Parallel

2
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)
3
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.

4
Reminders from Section 1

• We will use these same theorems to prove the
  lines are parallel given certain angle
  information.

5
Corresponding Angle Theorem

• If two parallel lines are cut by a transversal,
  then corresponding angles are congruent.
• // lines ? corresponding ?s are ?

6
Corresponding Angle Theorem
7
Alternate Interior Angle Theorem

• If two parallel lines are cut by a transversal,
  then alternate interior angles are congruent.
• // lines ? Alt. Int. ?s are ?

8
Alternate Interior Angle Theorem
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Alternate Exterior Angle Theorem

• If two parallel lines are cut by a transversal,
  then alternate exterior angles are congruent.
• // lines ? Alt. Ext. ?s are ?

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Alternate Exterior Angle Theorem
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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.

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Consecutive Interior Angle Theorem
1
2
m?1 m?2 180
13
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.

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Lesson 3-5 Postulates
Animation Construct a Parallel Line Through a
Point not on Line
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Lesson 3-5 Theorems
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Lesson 3-5 Example 1
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
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Lesson 3-5 CYP 1
Determine which lines, if any are parallel.I. e
fII. e gIII. f g

1. A
2. B
3. C
4. D

I only II only III only I, II, and III
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Lesson 3-5 Example 2
Solve Problems with Parallel Lines
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Lesson 3-5 Example 2
If Alt. Ext. angles are ?, then the lines will
be //
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.
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Lesson 3-5 Example 2
Solve Problems with Parallel Lines
7(15) 35 x 15 140 Simplify.
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Lesson 3-5 CYP 2

1. A
2. B
3. C
4. D

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Lesson 3-5 Example 3
Prove Lines Parallel
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Lesson 3-5 Example 3
Prove Lines Parallel
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Lesson 3-5 CYP 3
Given x y and , can you use
theCorresponding Angles Postulate to prove a
b?

1. A
2. B
3. C

yes no not enough informationto determine
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Lesson 3-5 Example 4
Slope and Parallel Lines

• Determine whether p q.

slope of p
slope of q
Answer Since the slopes are equal, p q.
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Lesson 3-5 CYP 4
Determine whether r s.

1. A
2. B
3. C

Yes, r is parallel to s. No, r is not parallel to
s. It cannot be determined.
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Homework

• Chapter 3-5
• Pg 175
• 1 5, 7 19, 23 (proof), 24(proof), 37, 50 52