# The Diagonals of an Isosceles Trapezoid Are Congruent

An isosceles trapezoid is a trapezoid whose sides are congruent.  An example of an isosceles trapezoid is shown below. The trapezoid $ABCD$ is isosceles with $AB$ parallel to $CD$ and $AD$ congruent to $BC$.

In this post, we are going to show that the diagonals of an isosceles trapezoid are congruent. In the figure below, we will show that $AC$ is congruent to $BD$.

Given:

Trapezoid $ABCD$ with $AB \parallel CD$.

What to show:  $AC \cong BD$.

Proof:

It is given that $ABCD$ is an isosceles trapezoid with $AB \parallel CD$.

By the definition of isosceles trapezoid $AD \cong BC$.

Now, since the base angles of an isosceles trapezoid  $\angle ADC \cong \angle BCD$.

Also, $CD \cong CD$, since congruence of segments is reflexive.

By the SAS Congruence postulate, $\triangle ADC \cong BCD$.

$AC \cong BD$ since corresponding parts of congruent triangles are congruent.

This completes the proof.

Therefore, diagonals of an isosceles trapezoid are congruent.