Download 2.8 Parallel Lines Cut By A Transversal

2.8 Parallel Lines Cut By A Transversal
When we extended the sides of triangles to find interior and exterior angle measurements, we created a
situation that was very similar to parallel lines cut by a transversal. In this section we will explore what types of
angles are created when we have a transversal cutting parallel lines.
Reviewing Vocabulary
Parallel lines in two dimensions are lines that never cross or intersect. This means that the lines have the
same orientation. If one line is going straight up and down, the parallel line will also be going straight up and down.
For our purposes we will only look at parallel lines that are not overlapping. In other words, one line will not be
sitting right on top of the other. Instead, our parallel lines will be more like railroad tracks.
A transversal is a line that intersects one or more parallel lines. This means that the transversal will have a
different orientation from the parallel lines. So if the parallel lines are going straight up and down, then the
transversal might be going left or right. The transversal could also be at some other angle (think of a positive 2
slope for example).
k
n
m
Line m is a transversal.
Lines n and k are parallel lines.
Notice that in this picture there are eight angles that are created. We typically name those angles using the
numbers 1 through 8. It could look something like this.
k
n
m
∠5
∠6
∠1
∠2
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∠3
∠4
∠7
∠8
You should notice right away that several
of these angles look like they have the same angle
measurement. In fact it looks like the four acute
angles have equal measurement and the four
obtuse angles have equal measurement. In fact
this is the case, but let’s examine why this is true
and classify the different types of angles we find
here.
Vertical Angles
Note that ∠1 and ∠3 must add up to 180° because they sit on a line. They are like the exterior and interior
angles of a triangle adding up to 180°. However, the same argument can be made for ∠4 and ∠3. They must also
add up to 180°. Therefore we know that ∠4 and ∠1 must have the same measurement. In other words we know
that ∠4 ≅ ∠1 (pronounced “angle 4 is congruent to angle 1”) or ∠4 = ∠1 (pronounced “the measure of angle
4 is equal to the measure of angle 1”).
k
n
m
∠5
∠6
∠1
∠2
∠7
∠8
∠3
∠4
We call this type of congruent angle vertical angles. One
way to remember vertical angles is to remember that they sit in a
“V”. Where are the other vertical angles in our picture?
The vertical angles come in pairs. A second pair of vertical
angles is ∠2 and ∠3. A third pair of vertical angles is ∠5 and ∠8.
The fourth pair of vertical angles is ∠6 and ∠7.
We can now ask questions such as: what is ∠3 if ∠2 =
40°? Since we know that they are vertical angles, they must be
congruent. Therefore the answer is ∠2 = 40°.
Corresponding Angles
Now imagine taking the angles formed by line n and line m and sliding them up so that they overlap the
angles formed by line k and line m. Now which angles do we know are congruent? In other words, angles 1 through
4 will be sitting right on top of angles 5 through 8. Which ones line up? These angles are called corresponding
angles and are congruent.
For starters, you should notice that ∠1 sits on top of ∠5. So ∠1 and ∠5 are a pair of corresponding angles.
A second pair would be ∠2 and ∠6. A third pair would be ∠3 and ∠7. The final pair would be ∠4 and ∠8. This
means that if ∠2 = 40° then ∠6 = 40° must be true since they are corresponding angles. One way to
remember corresponding angles is to think of the angles that are on the same corner. Cor-ner and Cor-responding
angles.
Alternate Interior Angles
Using the same picture above, look at ∠3 and ∠6. These are called alternate interior angles and are also
congruent. They are called alternate interior angles because they alternate which side of the transversal they are
on (∠3 is on top of the transversal in this case and ∠6 is on bottom) and because they are inside the parallel lines
(hence the word “interior”). Alternate interior angles are also congruent. There are two pairs of this type of angle
in our picture: ∠3 and ∠6 and then ∠4 and ∠5.
Alternate Exterior Angles
In a similar fashion, if the angles lie on alternate sides of the transversal and are outside the parallel lines,
the angles are called alternate exterior angles. Alternate exterior angles are congruent as well. In our picture there
are two pairs of alternate exterior angles which are ∠1 and ∠8 and then ∠2 and ∠7.
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Lesson 2.8
Use the following picture to answer the questions.
∠1 ∠5
∠2 ∠6
∠3 ∠7
∠4 ∠8
1. Name a pair of vertical angles.
2. Name a pair of corresponding angles.
3. Name a pair of alternate interior angles.
4. Name a pair of alternate exterior angles.
5. If ∠2 = 110°, what is ∠5?
6. If ∠2 = 110°, what is ∠4?
7. If ∠2 = 110°, what is ∠7?
8. If ∠1 = 70°, what is ∠8?
9. If ∠1 = 70°, what is ∠7?
10. If ∠2 = 140°, what are the measures of all the other angles?
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∠2
∠1
∠3 ∠4
∠6 ∠5
∠7 ∠8
11. Name all the pairs of vertical angles.
12. Name all the pairs of corresponding angles.
13. Name all the pairs of alternate interior angles.
14. Name all the pairs of alternate exterior angles.
15. If ∠2 = 40°, what is ∠8?
16. If ∠2 = 40°, what is ∠4?
17. If ∠1 = 140°, what is ∠7?
18. If ∠1 = 140°, what is ∠5?
19. If ∠1 = 140°, what is ∠6?
20. If ∠2 = 35°, what are the measures of all the other angles?
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