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DOM 3.1
Name the type of Angles
Ch 3.3
Parallel Lines &
Transversals
Warm-Up: Give a Reason
(a) Given: 1 & 2 are vertical s
Conclusion: 1  2
If vertical s, then 
(b) Given: 1  2, 2  3
Conclusion: 1  3
(c)
(d)
If A  B and B  C, then A  C
Given: A and B are a linear pair
Conclusion: A and B are supplementary
If lin pair, then supp.
Given: A and B are supplementary
Conclusion: mA + mB = 180°
If supp, then ms sum to 180°
Parallel Lines and Transversals
What would you call two lines which do not intersect?
Parallel
Exterior
B
A
Interior
D
A solid arrow placed
on two lines of a
diagram indicate the
lines are parallel.
C
Exterior
The symbol || is used to
indicate parallel lines.
AB || CD
Parallel Lines and Transversals
A slash through the parallel symbol || indicates the
lines are not parallel.
B
AB || CD
A
D
C
INTERIOR
–The space INSIDE the 2 lines
interior
EXTERIOR
-The space OUTSIDE the 2 lines
exterior
exterior
Special Angle Relationships
Interior Angles
Exterior
1
3 4
Interior
5 6
7 8
Exterior
2
<3 & <6 are Alternate Interior angles
<4 & <5 are Alternate Interior angles
<3 & <5 are Same Side Interior angles
<4 & <6 are Same Side Interior angles
Exterior Angles
<1 & <8 are Alternate Exterior angles
<2 & <7 are Alternate Exterior angles
<1 & <7 are Same Side Exterior angles
<2 & <8 are Same Side Exterior angles
Special Angle Relationships
WHEN THE LINES ARE
PARALLEL
♥Theorem 3.4 Alternate Interior
Exterior
1
2
3 4
Interior
5 6
7 8
Exterior
If the lines are not
parallel, these angle
relationships DO
NOT EXIST.
Angles are CONGRUENT
♥Theorem 3.6 Alternate Exterior
Angles are CONGRUENT
♥Theorem 3.5 Same Side Interior
Angles are SUPPLEMENTARY
♥Postulate 15 Corresponding
Angles are CONGRUENT
Theorem 3.7 Perpendicular
Transversal
• If a transversal is perpendicular to one of
the two parallel lines, then it is
perpendicular to the other.
j
h
k
jk
Corresponding Angles &
Consecutive Angles
Corresponding Angles: Two angles that occupy
corresponding positions.
 2   6,  1   5,  3   7,  4   8
Exterior
1
3
5 6
7 8
2
4
Interior
Exterior
Corresponding Angles
When two parallel lines are cut by a transversal, pairs of
corresponding angles are formed.
L
Line L
GPB = PQE
G
A
D
P
B
Q
E
F
Line M
Line N
GPA = PQD
BPQ = EQF
APQ = DQF
Four pairs of corresponding angles are formed.
Corresponding pairs of angles are congruent.
Same Side
Interior/Exterior Angles
Same Side Interior Angles: Two angles that lie between parallel
lines on the same sides of the transversal.
m3 +m5 = 180º, m4 +m6 = 180º
Same Side Exterior Angles: Two angles that lie outside parallel
lines on the same sides of the transversal.
Exterior
1 2
m1 +m7 = 180º, m2 +m8 = 180º
3 4
Interior
5 6
7 8
Exterior
Interior Angles
The angles that lie in the area between the two parallel lines
that are cut by a transversal, are called interior angles.
L
G
A 120
Line L
Exterior
0P
1200
600
D
B
600
F
Line N
APQ + DQP = 1800
Interior
E
Q
Line M
BPQ + EQP = 1800
Exterior
interior
eachside
pairofadd
AThe
pairmeasures
of interiorofangles
lieangles
on the in
same
theup to 1800.
transversal.
Alternate Interior/Exterior Angles
• Alternate Interior Angles: Two angles that lie between
parallel lines on opposite sides of the transversal (but not a
 3   6,  4   5
linear pair).
Alternate Exterior Angles: Two angles that lie outside parallel
lines on opposite sides of the transversal.
 2   7,  1   8
1 2 Exterior
3 4
Interior
5
7
6
8
Exterior
Alternate Interior Angles
Alternate angles are formed on opposite sides of the
transversal and at different intersecting points.
L
G
A
D
Line L
P
B
Q
E
Line M
Line N
BPQ = DQP
APQ = EQP
F
Two pairs of alternate angles are formed.
Pairs of alternate angles are congruent.
Let’s Practice
120°
1
3
60°
120°5
2 60°
4
120°
6 60°
7 8 120°
60°
m<1=120°
Find all the remaining
angle measures.
Ex. 4: Using properties of parallel lines
• Use the properties of parallel lines to find
the value of x.
120°
(x +15)°
Another practice problem
40
60°°
80°
100°
80°
80° 100°
40
180-(40+60)= 80°
60°
120
°
120° 60°
60°
Find all the missing
angle measures,
and name the
postulate or
theorem that
gives us
permission to
make our
statements.
Example 2: Complete the Proof
Given: p || q; m1 = 32°
Prove: m2 = 32°
Statement Reason
p || q; m1 = 32° Given
if parallel, then
2  1
corresp. s 
m2 = m1 if , then =
m2 = 32° if a = b, then subst a for b
Example 1: Proving the Alternate
Interior Angles Theorem
• Given: p ║ q
• Prove: 1 ≅ 2
1
2
3
Proof
Statements:
1. p ║ q
2. 1 ≅ 3
3. 3 ≅ 2
4. 1 ≅ 2
Reasons:
1. Given
2. Corresponding
Angles Postulate
3. Vertical Angles
Theorem
4. Transitive Property
of Congruence
Example 3: Solve for x
Given: p || q
Statement
p || q
1  2
2
p
Prove: x = 63
Reason
Given
q
1
If parallel, then alt. ext. s 
m1 = m2
If , then =
x – 8 = 55°
If a = b, then subst a for b
x = 63
If a = b, then a + c = b + c
Example 2: Using properties of parallel
lines
• Given that m 5 = 65°, find each measure.
Tell which postulate or theorem you use.
• A. m 6
B. m 7
• C. m 8
D. m 9
9
6
5
7
8
Solutions:
a. m 6 = m 5 = 65°
•
Vertical Angles Theorem
b. m 7 = 180° - m 5 =115°
•
Linear Pair postulate
c. m 8 = m 5 = 65°
•
Corresponding Angles Postulate
d. m 9 = m 7 = 115°
•
Alternate Exterior Angles Theorem
Closure
• Given that m3 = 55, what is m1?
Why?

2
3
1

Flag of
Puerto
Rico
Name the pairs of the following angles formed by a
transversal.
GG
G
AA
500
Line
Line
Line LL
L
P P
BBB
1300
D
DD
Q
Q
Q
EEE
FFF
Line
Line
Line
MM M
Line
Line
N
Line
NN
Assignment
#8-28 EVEN
#34-44 EVEN
(pg. 146-148)
