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.
Video: Definition of a Parallelogram
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.
By comparison, a quadrilateral with just one pair of parallel sides is a trapezoid in American English or a trapezium in British English.
The three-dimensional counterpart of a parallelogram is a parallelepiped.
Special cases
Quadrilaterals by symmetry
Rhomboid – A quadrilateral whose opposite sides are parallel and adjacent sides are unequal, and whose angles are not right angles
Rectangle – A parallelogram with four angles of equal size.
Rhombus – A parallelogram with four sides of equal length.
Square – A parallelogram with four sides of equal length and angles of equal size (right angles).
Video: What is a Parallelogram and what are its Special Cases?
Characterizations
A simple (non-self-intersecting) quadrilateral is a parallelogram if and only if any one of the following statements is true:
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 is 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.
The sum of the distances from any interior point to the sides is independent of the location of the point. (This is an extension of Viviani's theorem.)
There is a point X in the plane of the quadrilateral with the property that every straight line through X divides the quadrilateral into two regions of equal area.
Thus all parallelograms have all the properties listed above, and conversely, if just one of these statements is true in a simple quadrilateral, then it is a parallelogram.