# Parallelogram

A quadrilateral with both pairs of opposite sides parallel.
Try this Drag the orange dots on each vertex to reshape the parallelogram. Notice how the opposite sides remain parallel.

A parallelogram is a quadrilateral with opposite sides parallel. But there are various tests that can be applied to see if something is a parallelogram.

It is the "parent" of some other quadrilaterals, which are obtained by adding restrictions of various kinds:

• A rectangle is a parallelogram but with all four interior angles fixed at 90°
• A rhombus is a parallelogram but with all four sides equal in length
• A square is a parallelogram but with all sides equal in length and all interior angles 90°

## A quadrilateral is a parallelogram if:

1. Both pairs of opposite sides are parallel. (By definition). Or:
2. Both pairs of opposite sides are congruent. If they are congruent, they must also be parallel. Or:
3. One pair of opposite sides are congruent and parallel. Then, the other pair must also be parallel.

## Properties of a parallelogram

These facts and properties are true for parallelograms and the descendant shapes: square, rectangle and rhombus.
 Base Any side can be considered a base. Choose any one you like. If used to calculate the area (see below) the corresponding altitude must be used. In the figure above, one of the four possible bases and its corresponding altitude has been chosen. Altitude(height) The altitude (or height) of a parallelogram is the perpendicular distance from the base to the opposite side (which may have to be extended). In the figure above, the altitude corresponding to the base CD is shown. Area The area of a parallelogram can be found by multiplying a base by the corresponding altitude. See also Area of a Parallelogram Perimeter The distance around the parallelogram. The sum of its sides. See also Perimeter of a Parallelogram Opposite sides Opposite sides are congruent (equal in length) and parallel. As you reshape the parallelogram at the top of the page, note how the opposite sides are always the same length. Diagonals Each diagonal cuts the other diagonal into two equal parts, as in the diagram below. See Diagonals of a parallelogram for an interactive demonstration of this. Interior angles Opposite angles are equal as can be seen below. Consecutive angles are always supplementary (add to 180°) For more on both these properties, see Interior angles of a parallelogram.

## Parallelogram inscribed in any quadrilateral If you find the midpoints of each side of any quadrilateral, then link them sequentially with lines, the result is always a parallelogram.

This may seem counter-intuitive at first, but see Parallelogram inscribed in any quadrilateral for an animated exploration of this fact.