# 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 \$latex ABCD\$ is isosceles with \$latex AB\$ parallel to \$latex CD\$ and \$latex AD\$ congruent to \$latex 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 \$latex AC\$ is congruent to \$latex BD\$.

Given:

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

What to show:  \$latex AC \cong BD\$.

Proof:

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

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

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

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

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

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

This completes the proof.

Therefore, diagonals of an isosceles trapezoid are congruent.