Cut by a Transversal
Proving angles congruent
Some examples of corresponding angles are ∠1 and ∠5, ∠4 and ∠8, ∠7 and ∠3, and ∠6 and ∠2. These are corresponding because they are in the same place as the one they are paired with, and if you cut the transversal in half between the parallel lines, the angles would be the same. Thus, these are congruent angles.
Alternate Exterior Angles
Some examples of alternate exterior angles are ∠8 and ∠2, and ∠1 and ∠7. These are on opposite sides of the transversal (alternate) and outside of the parallel lines (exterior). These two facts bring us to the conclusion that these angles are alternate exterior angles. These are congruent angles.
Same Side Interior Angles
As the last type was self explanatory, so is this one. These angles are on the same side of the transversal, and inside the parallel lines. Basically, they are the opposite of the alternate exterior angles. Some examples are ∠4 and ∠5, and ∠3 and ∠6. These add to 180 degrees.
Same Side Exterior Angles
These angles are located on the same side of the transversal, but outside the parallel lines. They include ∠1 and ∠8, and ∠7 and ∠2. These add to 180 degrees.
Alternate Interior Angles
Alternate interior angles are angles which are on opposite sides of the transversal, but inside the parallel lines. For example, ∠4 and ∠6, or ∠5 and ∠3. As you can see from the examples, not only are they on opposite sides of the transversal, but they are also made with the opposite parallel line from each other. These types of angles are congruent.
Supplemental angles are angles which add to 180 degrees and are touching. For example, ∠1 is supplemental with ∠2 or with ∠4. ∠2 can be with ∠3 also. Each angle in a figure like this has two supplements.
Vertical angles are those which are opposite each other, but the vertex is the same point. In this diagram, these are ∠1 and ∠3, ∠2 and ∠4, ∠6 and ∠8, and ∠5 and ∠7.