Parallelogram

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In geometry, a parallelogram is a quadrilateral with two pairs of parallel sides. In Euclidean Geometry, 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.

Contents

Properties

  • Opposite sides of a parallelogram are equal in length.
  • Opposite angles of a parallelogram are equal in measure.
  • Adjacent angles are supplementary (add up to 180 degrees).
  • The area, A, of a parallelogram is A = bh, where b is the base of the parallelogram and h is its height.
  • Opposite sides of a parallelogram are parallel (by definition) and so will never intersect.
  • The area of a parallelogram is twice the area of a triangle created by one of its diagonals.
  • The area of a parallelogram is also equal to the magnitude of the vector cross product of two adjacent sides.
  • The diagonals of a parallelogram bisect each other.
  • Any non-degenerate affine transformation takes a parallelogram to another parallelogram.
    There is an infinite number of affine transformations which take any given parallelogram to a square.
  • A parallelogram has rotational symmetry of order 2 (through 180°). If it also has two lines of reflectional symmetry then it must be a rhombus or a rectangle.
  • The perimeter of a parallelogram is 2(a + b) where a and b are the lengths of adjacent

Types of parallelogram

  • 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 (right angles).
  • Rhombus - A parallelogram with four sides of equal length.
  • Square - A parallelogram with four sides of equal length and four angles of equal size (right angles).

Proof that diagonals bisect each other

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