Isoceles trapezoid tests

Dependencies: Parallelogram tests (F.04), Parallelogram properties (opposite sides equal, diagonals bisect each other) (F.03), Corresponding/alternate angles ⇔ lines parallel (converse of B.04), Isoceles converse: equal base angles ⇒ equal sides, SAS congruence.

Statement

Let $ABCD$ be a trapezoid with $AD \parallel BC$ (i.e. $AD$, $BC$ are the two bases, and $AB$, $CD$ are the two legs). The following three conditions are equivalent:

(a) Equal legs: $|AB| = |CD|$ (this is the definition of an isosceles trapezoid); (b) Equal same-base base angles: $\angle ABC = \angle DCB$ (the two base angles on the same base $BC$); (c) Equal diagonals: $|AC| = |BD|$.

Any one of these three may serve as the test for "$ABCD$ is an isosceles trapezoid".