Parallelogram tests

Dependencies: SSS congruence, SAS congruence, Equal alternate angles ⇒ parallel (B.02), Vertical angles are equal.

Statement

Let $ABCD$ be a quadrilateral (vertices labelled in order). The following four conditions are equivalent:

(a) Both pairs of opposite sides are parallel: $AB \parallel CD$ and $AD \parallel BC$ (this is the definition of a parallelogram); (b) Both pairs of opposite sides are equal: $|AB| = |CD|$ and $|AD| = |BC|$; (c) One pair of opposite sides is both parallel and equal: $AB \parallel CD$ and $|AB| = |CD|$; (d) Parallelogram: diagonals bisect each other: letting the diagonals $AC$ and $BD$ meet at $M$, we have $|AM| = |MC|$ and $|BM| = |MD|$.

Any one of the four can serve as a test for "$ABCD$ is a parallelogram".