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.

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 AC is congruent to BD.

diagons of an isosceles trapezoid

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.

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