Special parallelograms

The family. Rectangles and rhombuses are special parallelograms; a square is both. Like children in a family, each inherits everything from the parallelogram, plus extras of its own. Every rectangle is a parallelogram; not every parallelogram is a rectangle.

One right angle is enough. A parallelogram with one right angle is a rectangle: neighbouring angles add to 180°, so every angle is 90°. (In the same way, two neighbouring angles equal means each is 90°.)

Equal diagonals in a parallelogram ⇒ rectangle. SSS gives ∠A = ∠B, and they add to 180°.

A diagonal that bisects an angle ⇒ rhombus. If a diagonal of a parallelogram cuts an angle in half, all four sides are equal.

Two traps:

• every rhombus has perpendicular diagonals, so 'a rhombus with perpendicular diagonals is a square' is false;
• equal diagonals alone do not make a rectangle: an isosceles trapezium has equal diagonals too. Add 'and they bisect each other' and it becomes true.

Angle bisectors. The four angle bisectors of a parallelogram meet at right angles (½∠A + ½∠B = 90°), forming a rectangle, unless it is a rhombus, when they all meet at one point.