Parallelogram properties (opposite sides equal, diagonals bisect each other)

Dependencies: ASA congruence, Parallel ⇒ alternate angles equal (converse).

Statement

Let $ABCD$ be a parallelogram, with vertices labelled in either clockwise or counterclockwise order; by definition, the two pairs of opposite sides are parallel:

Then the lengths of these two pairs of opposite sides are equal as well:

Furthermore, letting the diagonals $AC$ and $BD$ meet at $M$, the two diagonals bisect each other — an immediate consequence within the same proof strategy (see "Immediate consequences" below).