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Concept
Parallelogram
Summary
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.
By comparison, a quadrilateral with at least 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.
The etymology (in Greek παραλληλ-όγραμμον, parallēl-ógrammon, a shape "of parallel lines") reflects the definition.
Rectangle – A parallelogram with four angles of equal size (right angles).
Rhombus – A parallelogram with four sides of equal length. Any parallelogram that is neither a rectangle nor a rhombus was traditionally called a rhomboid but this term is not used in modern mathematics.
Square – A parallelogram with four sides of equal length and angles of equal size (right angles).
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 parallel (by definition).
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.
Official source
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