Rectangle tests

Depends on: SSS congruence, Parallelogram properties (opposite sides equal, diagonals bisect each other), Parallelogram: opposite angles equal, the converse of Parallel ⇒ co-interior angles supp..

Statement

A rectangle is defined as "a parallelogram with one right angle" — that is, "parallelogram" is the default backdrop, and "one right angle" is the extra condition that upgrades the backdrop to a rectangle. Equivalently, the following three are mutually equivalent (and any of them can serve as a test for "$ABCD$ is a rectangle"):

(a) Three right angles: if at least three interior angles of quadrilateral $ABCD$ equal $90^\circ$, then it is a rectangle (the fourth angle is automatically $360^\circ - 270^\circ = 90^\circ$, and the converse of Parallel ⇒ co-interior angles supp. then gives opposite sides parallel, fitting the definition); (b) Parallelogram with equal diagonals: $ABCD$ is a parallelogram and $|AC| = |BD|$; (c) Parallelogram with one right angle: $ABCD$ is a parallelogram and $\angle A = 90^\circ$.

Among the three, (a) closes the rectangle directly from the angle side; (b) and (c) both take "already a parallelogram" as the starting point and add a single special condition (diagonal length, or one angle) to upgrade to a rectangle.