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 $latex BC$.

isosceles trapezoid

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 $latex BD$.

diagons of an isosceles trapezoid

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.

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