Exercise 32 Page 613

An isosceles triangle has two congruent sides and two congruent base angles. Additionally, an exterior angle is an angle that forms a linear pair with an interior angle of a triangle. Given this information, there are three possible cases we can draw.

Note that the first and third case shows exterior angles to two congruent angles. Therefore, we can reduce the number of cases to the first and the second.

First case

For the first case, due to the Linear Pair Postulate, we can write the measure of both base angles as:

Let's complete our diagram with this information.

Using the Triangle Sum Theorem, we can define the vertex angle as well:

Second case

In the second case, the exterior angle is adjacent to the vertex angle. By the Linear Pair Postulate, we can define the vertex angle as: Let's complete our diagram with this information. Notice that the base angles are congruent so by subtracting the expression of the vertex angle from $180^{\circ }$ and dividing by $2,$ we can write an expression for the base angles: