Consider a trapezoid with an upper base and a lower based. We can draw a straight line from each end of the top of the upper base and can make it perpendicular to the lower base. This creates two triangles, one on either side. 1. Since the line drawn represents the height, which is constant when not considering the slanted sides, these two lines are congruent. 2. These lines form a 90 degree angle with the base, so those two angles are also congruent. 3. If and only if the base angles are congruent, these triangles will be similar by AAS. 4. Since CPCTC, this means that if and only if the base angles are congruent, the two sides of the trapezoid will be congruent, making the trapezoid isosceles.