Parallelogram

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

parallelogram

∠A + ∠B = 180° and ∠A + ∠D = 180°
∠A = ∠C, ∠B = ∠D

Perimeter of a parallelogram:

P = 2a + 2b

Area of a parallelogram(rhombus):

parallelogram

$A = AB\cdot DE = BC \cdot DF$
$A = AB \cdot AD \sin \alpha$
$A = \frac12 AC \cdot BD \sin \gamma$

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

p2 + q2 = 2(a2 + b2)
or
AC2 + BD2 = 2(AB2 + BC2)

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