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 $latex BD$.
Trapezoid $latex ABCD$ with $latex AB \parallel CD$.
What to show: $latex AC \cong BD$.
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.