Introduction: Parallel lines
create situations in which some angles formed by intersecting lines (transversals) are equal. We will outline which angles are equal. The Lesson:
We use the diagram above where lines l and m are parallel and line t intersecting both l and m is called a transversal. We identify angles by their positions in this diagram. For example, angles 1 and 2 are both facing in the same direction, to the upper right. Such angles are called corresponding angles. Similarly we have angles 3 and 6, angles 4 and 7, and angles 8 and 5 as corresponding angles. Summarizing
Angles 8 and 2 and angles 3 and 7 are on opposite sides of the transversal and between (interior) the parallel lines. We call these angles alternate interior angles.
Angles 1 and 5 and angles 4 and 6 are on opposite sides of the transversal and above and below (exterior to) the parallel lines. We call these angles alternate exterior angles.
A direct result of the famous Parallel Postulate is that corresponding angles are equal. Accepting this fact gives us these relationships
Using these facts, especially the fact that corresponding angles are equal, we can show that other angles must also be equal.
Since angle 2 is supplementary to angle 6 and angle 1 is supplementary to angle 4, we know that angles 6 and 4 are equal because they are supplementary to equal angles. These angles 6 and 4 are alternate exterior angles.
Since angle 1 is supplementary to angle 3 and angle 7 is supplementary to angle 2 and angles 1 and 2 are equal, we also know that angles 7 and 3 are equal because they are supplementary to equal angles. These angles 7 and 3 are alternate interior angles.
: Let's Practice:
- Using the diagram below, suppose that angle 1 has a measure of 34º. What are the measures of the other angles in the diagram? You may assume that lines l and m are parallel.
Angle 8 is also 34º since it is vertical to angle 1.
Angle 2 has a measure of 34º since it is a corresponding angle to angle 1.
Angle 5 has a measure of 34º since it is vertical with angle 2, and also since it is an alternate exterior angle to angle 1.
The other angles are all supplementary to angles 1, 2, 5, and 8. Therefore they have a measure of 146º.
We could also conclude that angles 1 and 5 have the same measure since they are alternate exterior angles.
- In the diagram below, assume that lines l and m are parallel. Line t is a transversal. If angle 7 has a measure of 120º, what are the measures of the other angles?
Angle 6 is 120º since it is vertical to angle 7 and angle 3 is 120º since it is a corresponding angle with angle 7.
Angle 4 is 120º since it is vertical with angle 3 and angle 6 is 120º since it is an alternate exterior angle with angle 4.
All the other angles have a measure of 60 since, for example, angle 8 is supplementary to angle 3 and must add to 180 with angle 3.
This makes all the corresponding, vertical, and alternate exterior/interior angles 60º.