Download Lesson 3.3 Powerpoint - peacock

Angles Formed by
Transversals
Lesson 3.3
Objectives
• Identify angles formed by
transversals.
Key Vocabulary
•
•
•
•
•
•
•
Transversal
Interior angles
Exterior angles
Corresponding angles
Alternate interior angles
Alternate exterior angles
Same-side interior angles
Transversal
• Transversal - a line that intersects two or
more coplanar lines at different points.
Transversal
q
r
ℓ
Two coplanar lines
1 2
3 4
5 6
7 8
• Line ℓ is a transversal of lines q and r.
• Line ℓ forms a total of eight angles with
lines q and r. These angles, and specific
pairings of these angles, are given special
names.
Transversal
• Lines intersected by a transversal can be
parallel or not parallel.
ℓ
Transversal Angle Pair
Relationships
exterior
q
interior
r
1 2
3 4

Interior angles (4 ∠’s) lie in
the region between lines q
and r. (∠3, ∠4, ∠5, ∠6)

Exterior angles (4 ∠’s) lie in
the two regions that are not
between lines q and r, the
regions above line q and
below line r. (∠1, ∠2, ∠7, ∠8)
5 6
7 8
exterior
ℓ
Transversal Angle Pair
Relationships
exterior
q
interior
r
1 2
3 4

Same-side interior angles
– pair of interior angles that
lie on the same side of
transversal ℓ. (∠3 & ∠5, ∠4
& ∠6)

Alternate interior angles –
pair of nonadjacent interior
angles that lie on opposite
sides of transversal ℓ. (∠3 &
∠6, ∠4 & ∠5)
5 6
7 8
exterior
ℓ
Transversal Angle Pair
Relationships
exterior
q
interior
r
1 2

Alternate exterior angles –
pair of nonadjacent exterior
angles that lie on opposite
sides of transversal ℓ. (∠1 &
∠8, ∠2 & ∠7)

Corresponding angles –
pair of angles that lie on the
same side of transversal ℓ
and on the same side of
lines q and r. (∠1 & ∠5, ∠2 &
∠6, ∠3 & ∠7, ∠4 & ∠8)
3 4
5 6
7 8
exterior
Check For Understanding
Pairs Of Angles Formed by a Transversal
A line that intersects two or more lines at different points is
called a transversal.
G
A
C
Line L (transversal)
P
B
Line M
Line N
Q
D
F
FourLine
angles
are formed
atMpoint
Pparallel
and
four
point
Line
M and
line
N
areline
lines.P
L intersects
line
and
N another
at point
andatQ.
Eight angles are formed in all by the transversal L.
Q by the transversal L.
Corresponding Angles

A transversal creates two groups of four
angles in each group. Corresponding angles
are two angles, one in each group, in the
same relative position.
1 and 5 are corresponding
2 and 6 are corresponding
3 and 7 are corresponding
4 and 8 are corresponding
1 2
m
3 4
5 6
7 8
n
Alternate Interior Angles

When a transversal cuts two lines, alternate
interior angles are angles within the two lines
on alternate sides of the transversal.
1 and 2 are alternate interior angles
3 and 4 are alternate interior angles
m
1 3
4 2
n
Alternate Exterior Angles

When a transversal cuts two lines, alternate
exterior angles are angles outside of the two
lines on alternate sides of the transversal.
1 and 2 are alternate exterior angles
1 3
m
3 and 4 are alternate exterior angles
4 2
n
Same-Side Interior Angles

When a transversal cuts two lines, interior
angles on the same side of the transversal
are angles within the two lines on the same
side of the transversal.
1 and 2 are interior angles on
the same side of the transversal
3 and 4 are interior angles on
the same side of the transversal
m
1 3
2 4
n
Name the pairs of the following angles formed by a
transversal.
GG
G
AA
D
DD
Line
Line
Line LL
L
P P
BBB
Q
Q
Q
EEE
FFF
Line
Line
Line
MM M
Line
Line
N
Line
NN
Parallel Lines and Transversals
Determine if the statement is true or false. If false,
correct the statement.
1 2
4 3
5 6
8 7
r
9 10
12 11
13 14
16 15
p
q
s
1. Line r is a transversal of
lines p and q.
True – Line r intersects both
lines in a plane.
2. ∠2 and ∠6 are alternate
interior angles.
False - The angles are
corresponding angles
on transversal p.
Parallel Lines and Transversals
Determine if the statement is true or false. If false,
correct the statement.
3. ∠9 and ∠11 are alternate
interior angles.
1 2
4 3
5 6
8 7
r
9 10
12 11
13 14
16 15
p
q
s
False – The angles are
vertical angles created by
the intersection of q and r.
4. ∠1 and ∠7 are alternate
exterior angles.
True - The angles are
alternate exterior angles
on transversal p.
Parallel Lines and Transversals
Determine if the statement is true or false. If false,
correct the statement.
5. ∠12 and ∠14 are
alternate interior angles.
1 2
4 3
5 6
8 7
r
9 10
12 11
13 14
16 15
p
q
s
True – The angles are
alternate interior angles
on transversal q.
6. ∠6 and ∠13 are sameside interior angles.
True – The angles are
same-side interior angles
on transversal s.
Parallel Lines and Transversals
Determine if the statement is true or false. If false,
correct the statement.
7. ∠9 and ∠10 are alternate
exterior angles.
1 2
4 3
5 6
8 7
r
9 10
12 11
13 14
16 15
p
q
s
False – The angles are a
linear pair with linear rays
on line r.
8. ∠8 and ∠16 are
corresponding angles.
True – The angles are
corresponding on
transversal s.
Example 1
Describe the relationship between the angles.
a. 1 and 2
b. 3 and 4
SOLUTION
a. alternate interior angles
b. alternate exterior angles
c. same-side interior angles
c. 5 and 6
Example 2
List all pairs of angles that fit the
description.
a. corresponding
b. alternate exterior
c. alternate interior
d. same-side interior
SOLUTION
a. corresponding
1 and 5
3 and 7
2 and 6
4 and 8
Example 2
b. alternate exterior:
1 and 8
3 and 6
c. alternate interior:
2 and 7
4 and 5
d. same-side interior:
2 and 5
4 and 7
Your Turn:
Describe the relationship between the angles.
1.
2 and 7
ANSWER
2.
alternate exterior angles
3 and 5
ANSWER
3.
same-side interior angles
1 and 5
ANSWER
corresponding angles
4.
4 and 5
ANSWER
5.
alternate interior angles
4 and 8
ANSWER
6.
corresponding angles
4 and 6
ANSWER
same-side interior angles
Example 3a:
Identify ∠7 and ∠3 as alternate interior, alternate
exterior, corresponding, or same-side interior
angles.
Answer: corresponding
Example 3b:
Identify ∠8 and ∠2 as alternate interior, alternate
exterior, corresponding, or same-side interior
angles.
Answer: alternate exterior
Example 3c:
Identify ∠5 and ∠1 as alternate interior, alternate
exterior, corresponding, or same-side interior
angles.
Answer: corresponding
Example 3d:
Identify ∠7 and ∠1 as alternate interior, alternate
exterior, corresponding, or same-side interior
angles.
Answer: alternate exterior
Example 3e:
Identify ∠3 and ∠9 as alternate interior, alternate
exterior, corresponding, or same-side interior
angles.
Answer: alternate interior
Example 3f:
Identify ∠7 and ∠10 as alternate interior, alternate
exterior, corresponding, or same-side interior
angles.
Answer: same-side interior
Your Turn:
Identify each pair of angles as alternate interior,
alternate exterior, corresponding, or same-side
interior angles.
a.
Answer: same-side interior
b.
Answer: corresponding
c.
Answer: alternate exterior
Your Turn:
Identify each pair of angles as alternate interior,
alternate exterior, corresponding, or same-side
interior angles.
d.
Answer: alternate interior
e.
Answer: corresponding
f.
Answer: alternate exterior
Helpful Hint
To determine which line is the transversal
for a given angle pair, locate the line that
connects the vertices.
Example 4: Identifying Angle Pairs and
Transversals
Identify the transversal and classify each angle pair.
A. 1 and 3
transversal l
corr. s
B. 2 and 6
transversal n
alt. int s
C. 4 and 6
transversal m
alt. ext s
Your Turn
Identify the transversal and classify the angle pair 2 and 5
in the diagram.
transversal n
consecutive int. s.
Joke Time
• What has 18 legs and catches flies?
• A baseball team.
• Who stole the soap?
• The robber ducky!
• What happened to the plant on the
windowsill of the classroom?
• It grew square roots!
Assignment
• Section 3.3, pg. 123-125: #1-8 all, 9-41 odd