Parallelogram

Dependencies: Quadrilateral, Parallel. A derived definition: require both pairs of opposite sides of a quadrilateral to be parallel.

Definition

Let $ABCD$ be a quadrilateral.

If both pairs of opposite sides are parallel, so that $AB\parallel DC$ and $AD\parallel BC$, then $ABCD$ is a parallelogram.

This one condition forces opposite sides and angles to be equal, adjacent angles to be supplementary, and diagonals to bisect each other. Parallelograms are therefore the common foundation of the special quadrilaterals.

Remarks

• One condition yields many properties. Opposite sides are equal, opposite angles are equal (Parallelogram: opposite angles equal), and diagonals bisect each other (Parallelogram: diagonals bisect each other); see Parallelogram properties (opposite sides equal, diagonals bisect each other).
• Several converse tests are available. A quadrilateral is a parallelogram if both pairs of opposite sides are equal, one pair is both equal and parallel, or the diagonals bisect each other (Parallelogram tests).
• Rectangles, rhombi, and squares are special cases. Add a right angle for a rectangle, or equal adjacent sides for a rhombus. Adding both gives a square.
• Central symmetry. A parallelogram is invariant under a $180^\circ$ rotation about the intersection of its diagonals.