Download 3.3 Parallel Lines & Transversals

Parallel Lines & Transversals 3.3
Transversal
A line, ray, or segment that intersects 2 or more
COPLANAR lines, rays, or segments.
Parallel
lines
Non-Parallel
lines
transversal
transversal
If two parallel lines are cut by a transversal,
then the pairs of corresponding angles
are congruent.
Corresponding Angles Postulate
2 1
3 4
6 5
7 8
1
5
22
66
33
77
44
88
If two parallel lines are cut by a transversal,
then the pairs of alternate interior angles
are congruent.
Alternate Interior Angles Postulate
2 1
3 4
6 5
7 8
3
5
4
6
If two parallel lines are cut by a transversal,
then the pairs of consecutive interior angles
are supplementary.
Consecutive Interior Angles Postulate
2 1
3 4
6 5
7 8
m 4 + m 5 = 180°
m 3 + m 6 = 180°
If two parallel lines are cut by a transversal,
then the pairs of alternate exterior angles
are congruent.
Alternate Exterior Angles Postulate
2 1
3 4
6 5
7 8
1
7
2
8
If a transversal is perpendicular to one of two parallel
lines, then it is perpendicular to the other.
Perpendicular Transversal Theorem
j
k
Prove the Alternate Interior Angles Theorem.
GIVEN
p || q
PROVE
1 2
Statements
1
p || q
Reasons
1
Given
2
1   3
2
Corresponding Angles Postulate
3
3  2
3
Vertical Angles Theorem
4
1  2
4
Transitive property of Congruence
Using Properties of Parallel Lines
Given that m5 = 65°,
find each measure. Tell
which postulate or theorem
you use.
m 6 = m 5 = 65°
m 7 = 180° – m 5 = 115°
Vertical Angles Theorem
Linear Pair Postulate
m
8 = m
5 = 65°
Corresponding Angles Postulate
m
9 = m
7 = 115°
Alternate Exterior Angles Theorem
PROPERTIES OF SPECIAL PAIRS OF ANGLES
Use properties of
parallel lines to find
the value of x.
m
m
4 = 125°
4 + (x + 15)° = 180°
125° + (x + 15)° = 180°
x = 40°
Corresponding Angles Postulate
Linear Pair Postulate
Substitute.
Subtract.
GIVE AN EXAMPLE OF EACH ANGLE PAIR
Give an example of each angle pair.
A. corresponding angles
1 and 5 or 2 and 6 or
4 and 8 or 3 and 7
B. alternate interior angles
3 and 5 or 4 and 6
C. alternate exterior angles
1 and 7 or 2 and 8
D. consecutive interior angles
3 and 6 or 4 and 5
GIVE AN EXAMPLE OF EACH ANGLE PAIR
A. corresponding angles
1 and 3
B. alternate interior angles
2 and 7
C. alternate exterior angles
1 and 8
D. consecutive interior angles
2 and 3
Special Angle Relationships
Interior Angles
1
3 4
5 6
7 8
2
3 & 6 are Alternate Interior angles
4 & 5 are Alternate Interior angles
3 & 5 are Consecutive Interior angles
4 & 6 are Consecutive Interior angles
Exterior Angles
1 & 8 are Alternate Exterior angles
2 & 7 are Alternate Exterior angles
1 & 7 are Consecutive Exterior angles
2 & 8 are Consecutive Exterior angles
Special Angle Relationships
WHEN THE LINES ARE PARALLEL
1
3
5
2
♥Alternate Interior Angles are
CONGRUENT
4
6
7 8
♥Alternate Exterior Angles are
CONGRUENT
♥Consecutive Interior Angles are
SUPPLEMENTARY
If the lines are not
parallel, these
angle relationships
DO NOT EXIST.
♥ Corresponding Angles are
CONGRUENT
♥Consecutive Exterior Angles are
SUPPLEMENTARY
Let’s Practice
120°1
60°3
120° 5
60° 7
2 60°
4 120°
6 60°
8 120°
m1=120°
Find all the remaining
angle measures.
Find the value of x, name the angles.
a. x = 64
b. x = 75
c. x = 12
d. x = 40
e. x = 60
f. x = 60
g. x = 90
h. x = 15
i. x = 20
How would you show that the given lines are parallel?
43
a. a and b
Consecutive Interior
`s Supplementary
d. e and g
b. b and c
Corresponding
`s Congruent
e. a and c
Calculate the missing  Consecutive Interior
Corresponding
`s Supplementary
`s Congruent
c. d and f
Corresponding
`s Congruent
Find the value of each variable.
1. x x = 2
2. y y = 4
Find the value of x and y that make the lines parallel,
110
name the angles.
110
70
a. x
110
b. y
2x + 2 = x + 56
y + 7 = 70
x = 54
y = 63
2(54) + 2 = 110
2(63) – 16 = 110
Corresponding `s
Consecutive Exterior
Congruent
`s are Supplementary
IDENTIFY THE TRANSVERSAL, &
CLASSIFY EACH ANGLE PAIR
a. 2 and 16
q
p
1
2
8 7
3 4
6 5
Transversal p
Lines r and s
r
Alternate Exterior ’s
b. 6 and 7
Transversal r
Lines p and q
9 10
11 12
16 15 14 13
Consecutive Interior ’s
s
IDENTIFY THE TRANSVERSAL, &
CLASSIFY EACH ANGLE PAIR
A. 1 and 3
transversal l
corresponding s
B. 2 and 6
transversal n
alternate interior s
C. 4 and 6
transversal m
alternate exterior s
Review
If two lines are intersected by a transversal
and any of the angle pairs shown below are
congruent, then the lines are parallel. This
fact is used in the construction of parallel
lines.
Assignment
3.3A and 3.3B
Section 9 - 33