Using congruence to establish a test for rhombuses

Theorem

A quadrilateral whose diagonals bisect each other at right angles is a rhombus.

Proof

Let \(ABCD\) be a quadrilateral whose diagonals bisect each other at right angles at M.

We prove that \(DA = AB\). It follows similarly that

\(AB = BC\ \text{and} \ BC = CD\)

\(\triangle ABC \equiv \triangle CDA\) (AAS)

So \(AB = AD\) and by the first test above \(ABCD\) is a rhombus.