Download Parallel Lines with a Transversal

Parallel Lines
and a Transversal
Vocabulary
• Parallel lines: Lines in the same plane that have
the same slope and never intersect.
• Transversal: A line that intersects two or more
parallel lines.
Vocabulary
• Interior Angles: Angles that lie between
the parallel lines.
• Same Side Interior Angles: Interior angles
on the same side of the transversal.
• Alternate Interior Angles: Angles that lie
between the parallel lines and on opposite
sides of the transversal, and are
congruent.
Vocabulary
• Exterior Angles: Angles that lie outside the
parallel lines.
• Same Side Exterior Angles: Exterior angles
on the same side of the transversal.
• Alternate Exterior Angles: Angles that lie
outside the parallel lines and on opposite
sides of the transversal, and are
congruent.
Vocabulary
• Corresponding Angles: Angles that are in
the same relative position and are
congruent.
• Supplementary Angles: Angles that have a
sum of 180.
• Linear Pair: Two angles with a common
side that are supplementary.
• Vertical Angles: Formed when two lines
cross, they are on opposite sides of both
lines.
Theorem
• When parallel lines are cut by a
transversal, then the pairs of
corresponding angles are congruent,
the pairs of alternate interior angles
are congruent, and the pairs of
alternate exterior angles are
congruent.
Converse of the Theorem
• When corresponding angles or
alternate interior angles or
alternate exterior angles are
congruent, then the lines that are
cut by the transversal to form the
angles are parallel.
Corresponding Angles
Observe where the angles from the top intersection end up
when it is “cut out” and slid down the transversal…
Identifying Angles.
In the figure below, 𝐿1 ∥ 𝐿2 , and 𝑚 is a transversal.
Parallel lines:
Transversal:
Interior Angles:
Exterior Angles:
Corresponding Angles:
Alt. Int. Angles:
Alt. Ext. Angles:
Supplementary:
Vertical Angles:
Linear Pairs:
Discussion
Consider the relationships between the angles…
How many angles do we need to know the measure of in order
to determine the measure of the rest? 1
How many different angle measures should we expect to see
amongst the angles? 2
What if…
What if L1 and L2 were not parallel?
L1
2
1
3
4
L2
6
5
7
8
m
Which angles are corresponding angles? Are they
congruent? Why or why not?
Example 1
𝑘
𝑙
𝑚
1
5
2
3
6
7
𝑛
9
13
8
10
11
14
4
12
15
16
1. If ∠1 ≅ ∠3 is k parallel to l? Be prepared to Explain.
y: yes
n: no
2. If ∠9 ≅ ∠6 is m parallel to n? Be prepared to Explain.
y: yes
n: no
Example 2
Using the diagram below, determine if angles 1 and 9 are:
a) interior b) exterior c) vertical d) corresponding
𝑘
𝑙
𝑚
1
5
2
3
6
7
𝑛
9
13
10
11
14
12
15
16
4
8
Example 3
Using the diagram below, determine if angles 10 and 15 are:
a) Alternate Interior b) Alternate Exterior c) Vertical
d) Corresponding
𝑘
𝑙
𝑚
1
5
2
3
6
7
𝑛
9
13
10
11
14
12
15
16
4
8
Example 4
Using the diagram below, determine if angles 1 and 4 are:
a) interior b) exterior c) vertical d) corresponding
𝑘
𝑙
𝑚
1
5
2
3
6
7
𝑛
9
13
10
11
14
12
15
16
4
8
Example 5
Using the diagram below, determine if angles 5 and 2 are:
a) interior b) exterior c) vertical d) corresponding
𝑘
𝑙
𝑚
1
5
2
3
6
7
𝑛
9
13
10
11
14
12
15
16
4
8