Parallelogram

Parallelogram

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.

Video: Properties of a Parallelogram

Other properties

  • Opposite sides of a parallelogram .....

 

 

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