# Proof that Isosceles Trapezoids Have Congruent Base Angles

##### Overview

A trapezoid is a quadrilateral with exactly one pair of parallel sides (definitions may differ in other sources). The parallel sides are the base and the non parallel sides are called the legs. An isosceles trapezoid is a trapezoid whose base are congruent.

In this post, we prove that  if a trapezoid is isosceles, its base angles are congruent. In trapezoid ABCD below, $\overline{BC} \parallel \overline{AD}$ and $\overline{AB} \cong \overline{CD}$

In the proof below, we need to show that $\angle A \cong \angle B$.

##### The Isosceles Trapezoid Theorem

The base angles of an isosceles trapezoid are congruent.

##### Proof

Draw two perpendiculars to the base $\overline{BE}$ and $\overline{CF}$ (S).

$\overline{BE} \cong \overline{CF}$ since they are altitudes of the same trapezoid(S).

$\overline{AB} \cong \overline{CD}$ (Given)(S)

$\overline{AE} \cong \overline {FD}$ by the Hypotenuse Leg Theorem.

$\triangle ABC \cong \triangle DCF$ (SSS Triangle Congruence)

$\angle A \cong \angle D$ since corresponding parts of congruent triangles are congruent.

## 2 thoughts on “Proof that Isosceles Trapezoids Have Congruent Base Angles”

1. Before using HL, you should point out the triangles are right because the altitudes form perpendiculars with the base. The Hypotenuse Leg Theorem says that triangle BAE is congruent to triangle CDF (since you’ve shown the hypotenuses and a leg are congruent)–NOT that the other legs are congruent. Once you know the triangles are congruent, the base angles are congruent by CPCTC. Your HL step is incorrect and your SSS step is not needed.