In Euclidean geometry, a parallelogram is a simple (non self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct
consequence of the Euclidean Parallel Postulate and neither condition can be proven without appealing to the Euclidean Parallel Postulate or one of its equivalent formulations. The three-dimensional
counterpart of a parallelogram is a parallelepiped.
The etymology (in Greek παραλληλ-όγραμμον, a shape "of parallel lines") reflects
the definition.
A simple (non self-intersecting) quadrilateral is a parallelogram if and only if any one of
the following statements is true:12
Two pairs of opposite sides are equal in length.
Two pairs of opposite angles are equal in measure.
The diagonals bisect each other.
One
pair of opposite sides are parallel and equal in length.
Adjacent angles are supplementary.
Each diagonal divides the quadrilateral into two congruent triangles.
The sum of the squares of the sides equals the sum of the squares of the diagonals. (This is the parallelogram law.)
It has rotational symmetry of order 2.
