Parallelogram
Parallelogram
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
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Rhomboid – A quadrilateral whose opposite sides are parallel and adjacent sides are unequal, and whose angles are not right angles
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Rectangle – A parallelogram with four angles of equal size.
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Rhombus – A parallelogram with four sides of equal length.
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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:
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Two pairs of opposite sides are equal in length.
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Two pairs of opposite angles are equal in measure.
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The diagonals bisect each other.
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One pair of opposite sides is parallel and equal in length.
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Adjacent angles are supplementary.
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Each diagonal divides the quadrilateral into two congruent triangles.
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The sum of the squares of the sides equals the sum of the squares of the diagonals. (This is the parallelogram law.)
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It has rotational symmetry of order 2.
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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.)
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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.
Video: Properties of a Parallelogram
Other properties
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Opposite sides of a parallelogram .....
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