Sec. 32 Proving Parallel Lines

1
Sec. 3-2Proving Parallel Lines

• Objective
• To use a Transversal in Proving Lines Parallel.
• To relate Parallel Perpendicular Lines.

2
Now we will work the proofs backwards.

• In the last section we started with // lines and
  worked toward the angles.
• In this section we will start with the angles and
  work towards the // lines.

3
P(3-2) Converse of the Corresponding Angle
Theorem

• If two lines a transversal intersect to form
  corresponding angles that are congruent then the
  two lines are //.

1
m
If ?1 ? ?2, then m // n
2
n
4
Th(3-3) Converse of the Alternate Interior Angle
Theorem

• If two lines a transversal intersect to form
  Alternate Interior that are congruent then the
  two lines are //.

?3 ? ?6 ?4 ? ?5
3
4
5
6
5
Proof
1
2
n
Given ?3 ? ?6 Prove n // m
4
n
3
6
5
m
8
7

• Reasons
• Given
• 2. Vertical ?s are ?
• 3. Subs.
• 4. If corresp ?s
• are ? then
• lines are //.

• Statements
• ?3 ? ?6
• ?3 ? ?1
• 3. ?1 ? ?6
• 4. n // m

6
Th.(3-4) Converse of Same-Sided Interior Angle
Theorem.

• If two lines a transversal intersect to form
  same - sided interior angles that are
  supplementary then the two lines are //.

3
4
m?3 m?5 180
5
6
m?4 m?6 180
7
Proof
Given m?3 m?5 180 Prove n // m
n
3
5
7
m
Statements 1. m?3 m?5 180 2. m?5 m?7
180 3. m?3 m?5 m?5 m?7 4. m?3 m?7 5. ?3
? ?7 6. n // m

• Reason
• Given
• ? Add. Post.
• Subs.
• Subtr.
• 5. Def. of ?
• 6. If corrsp. ?s are ?, then lines are //.

8
Th(3-5) If two lines are // to the same line,
then they are // to each other.
t
1
2
k
3
4
5
6
m
7
8
9
10
n
11
12
9
Th(3-5) If two lines are // to the same line,
then they are // to each other.
t
k
5
6
m
7
8
9
10
n
11
12
10
Th(3-5) In a plane, if 2 lines are perpendicular
to the same line, then they are // to each other.
t
r
s
11
Corresponding Angles are ? They are 90
Alt. Int. ?s are ? They are 90
Same-sided int. ?s are Supplementary They are
both 90
t
r
s
12
Example 1 Solve for x and then solve for each
angle such that n // m.
14 3x 14 3(40) 134
n
14 3x
5x - 66
m

• 14 3x 5x -66
• -3x -3x
• 14 2x 66
• 66 66
• 80 2x
• 2
• 40 x

5x 66 5(40) 66 134
13
Example 2 Find the m?1
62
7x - 8
1
7x 8 7(18) 8 118
7x 8 62 180 7x 54 180 7x 126 x 18