GEO

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.

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.