Chapter 3-5
Proving Lines Parallel
• 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)
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.
p
q
r
Reminders from Section 1
We will use these same theorems to
prove the lines are parallel given
certain angle information.
Corresponding Angle Theorem
If two parallel lines are cut by a
transversal, then corresponding
angles are congruent.
// lines  corresponding s are 
Corresponding Angle Theorem
Alternate Interior Angle Theorem
If two parallel lines are cut by a
transversal, then alternate
interior angles are congruent.
// lines  Alt. Int. s are 
Alternate Interior Angle Theorem
Alternate Exterior Angle Theorem
If two parallel lines are cut by a
transversal, then alternate
exterior angles are congruent.
// lines  Alt. Ext. s are 
Alternate Exterior Angle Theorem
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.
Consecutive Interior Angle Theorem
1
2
m1 + m2 = 180
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.
p
m
n
Animation: Construct a Parallel Line
Through a Point not on Line
Identify Parallel Lines
Determine which lines,
if any, are parallel.
77o
Consec. Int. s are supp.
 a//b
Alt. Int. s are not 
 a is not // c
Consec. Int. s are not supp.
 b is not // c
Determine which lines,
if any are parallel.
I. e || f
II. e || g
III. f || g
A. I A
only
B. IIBonly
D. I,DII, and III
0%
0%
D
0%
C
A
0%
B
C. IIIConly
Solve Problems with Parallel Lines
ALGEBRA Find x and m ZYN so that
||
.
Explore From the figure, you know that
m WXP = 11x – 25 and m ZYN = 7x + 35.
You also know that WXP and ZYN are
alternate exterior angles.
ALGEBRA Find x and m ZYN so that
||
.
If Alt. Ext. angles are , then the lines will be //
m WXP = m ZYN
Alternate exterior  thm.
11x – 25 = 7x + 35
Substitution
4x – 25 = 35
4x = 60
x = 15
Subtract 7x from each side.
Add 25 to each side.
Divide each side by 4.
Solve Problems
with Parallel Lines
Now use the value of x to find m ZYN.
m ZYN = 7x + 35
Original equation
= 7(15) + 35
x = 15
= 140
Simplify.
Answer: x = 15, m ZYN = 140
ALGEBRA Find x so that
||
.
A. xA= 60
B. xB= 9
C. xC= 12
D. xD= 12
0%
D
0%
C
0%
B
A
0%
Prove Lines Parallel
Given: ℓ || m
Prove: r || s
Prove Lines Parallel
Proof:
Statements
1.
2.
3.
4.
5.
6.
7.
Reasons
1. Given
2. Consecutive Interior Angle
Theorem
3. Definition of supplementary
angles
4. Definition of congruent angles
5. Substitution
6. Definition of supplementary
angles
7. If consecutive interior angles
theorem
Given x || y and
, can you use the
Corresponding Angles Postulate to prove a || b?
A. yes
A
B. no
B
0%
0%
C
A
0%
B
C. not
C enough
information
to determine
Slope and Parallel Lines
Determine whether p || q.
slope of p:
slope of q:
Answer: Since the slopes are equal, p || q.
Determine whether r || s.
A. Yes,
A r is
parallel to s.
B.
B
No, r is not
parallel to s.
C. C
It cannot be
determined.
0%
C
0%
B
A
0%
Homework
Chapter 3-5
• Pg 175
1 – 5, 7 – 19, 23 (proof), 24(proof), 37, 50 – 52