Isosceles Trapezoid Diagonals Theorem

To prove a biconditional statement, the conditional statement and its converse must be proven. Start by assuming that a trapezoid is isosceles.

If a Trapezoid Is Isosceles, Then Its Diagonals Are Congruent

Let ABCD be an isosceles trapezoid with AB≅CD. By the Isosceles Trapezoid Base Angles Theorem, the base angles are congruent, that is, ∠ A ≅ ∠ D.

Next, draw the diagonals and separate the triangles ABD and DCA. The Reflexive Property of Congruence gives that AD≅ AD.

Notice that △ ABD ≅ △ DCA by the Side-Angle-Side (SAS) Congruence Theorem. As a result of that relationship, AC ≅ BD.

If the Diagonals of a Trapezoid Are Congruent, Then It Is Isosceles

For the converse, consider a trapezoid with congruent diagonals.

Next, draw a line parallel to BD passing through C and let P be intersection point between this line and AD.

Since AD∥BC and BD∥CP, BCPD is a parallelogram. Therefore, CP≅BD and then, by the Transitive Property of Congruence, CP≅ AC. This makes △ ACP an isosceles triangle.

The Isosceles Triangle Theorem leads to the conclusion that ∠ CAD ≅ ∠ CPA. Additionally, the Corresponding Angles Theorem indicates that ∠ CPA ≅ ∠ BDA. Next, separate triangles ABD and DCA.

By the Side-Angle-Side (SAS) Congruence Theorem, △ ABD ≅ △ DCA. This implies that AB≅CD which makes ABCD an isosceles trapezoid.