parallelograms

Parallelograms are a class of quadrilaterals characterized by two pairs of opposite sides that are parallel. This parallelism implies several key properties: opposite sides are equal in length; opposite angles are equal. The diagonals of a parallelogram bisect each other, meeting at a point that splits each diagonal into two equal parts. Consecutive interior angles are supplementary, so adjacent angles sum to 180 degrees. The parallelogram's area can be computed as base times height, A = b h, where h is the perpendicular distance between the bases; equivalently, if the lengths of two adjacent sides are a and b and the included angle is theta, A = a b sin theta. The diagonals divide the parallelogram into two pairs of congruent triangles.

Common special cases include rectangle, a parallelogram with all right angles; rhombus, a parallelogram with all sides equal; square, a figure that is both a rectangle and a rhombus. In a rhombus the diagonals are perpendicular and they bisect the vertex angles; in a rectangle the diagonals are equal in length.

Parallelograms appear in many contexts, including geometry, trigonometry, and computer graphics, and serve as a simple model for slanted quadrilaterals. They can be defined in coordinates by two vectors representing adjacent sides; the area equals the magnitude of their cross product, |u × v|.