# Parallelogram

**A quadrangle** with two pairs of parallel sides is called a **parallelogram**.

Each pair of opposite sides in a parallelogram are equal.

In order to determine whether a quadrangle is parallelogram, we will use the following criteria:

— if the two pairs of opposite sides in a quadrangle are equal, then this quadrangle is a parallelogram;

— if two opposite sides in a quadrangle are parallel and equal, then this quadrangle is a parallelogram;

— if, in a quadrangle, the diagonals bisect each other, then this quadrangle is a parallelogram;

A quadrangle which has four right angles is called a **rectangle**.

The following statement is valid: If a parallelogram is a rectangle, then its diagonals are equal in length.

Vice versa, if the diagonals of a parallelogram are equal in length, then this parallelogram is a rectangle.

A parallelogram that has two adjacent equal sides, is called a **rhombus**. The following statement about the rhombus is valid:

If a parallelogram is a rhombus, then its diagonals are perpendicular.

Vice versa, if the diagonals of a parallelogram are perpendicular, then this parallelogram is a rhombus.

**A square** may be considered as rectangle which has equal adjacent sides, or a rhombus with a right angle.

Therefore, a square has all the properties of a rectangle and a rhombus.

#### Parallelogram formulas

∠A = ∠C, ∠B = ∠D

*Perimeter of a parallelogram:*

*Area of a parallelogram(rhombus):*

$A = AB \cdot AD \sin \alpha$

$A = \frac12 AC \cdot BD \sin \gamma$

Relationship between the sides and diagonals of a parallelogram(rhombus):

^{2}+ q

^{2}= 2(a

^{2}+ b

^{2})

or

AC

^{2}+ BD

^{2}= 2(AB

^{2}+ BC

^{2})