Suppose you need to find the unknown angles in a complicated figure and figure out which side is the longest. Here is a figure with several unknown angles.

One way to start is to consider the equilateral triangle . Since all sides are congruent, all angles in that triangle are congruent. Since the angles must add up to 180 degrees, each angle in triangle is 60 degrees, because .

Next, consider the angles with a vertex at . Using rules for supplementary and vertical angles, we can find these unknown angles. Angle is supplementary to angle , so the measure of angle is 120 degrees because . Since angle forms a vertical pair with angle , these angles are congruent. Similarly, angle forms a vertical pair with angle , so the measures of angles  and are equal.

Finally, we can use the Triangle Angle Sum Theorem to find the two unknown angles. The measure of angle is 45 degrees, because , and the measure of angle is 25 degrees, because .