Parallelogram Basics

What is a Parallelogram?

Parallelogram is a quadrilateral with two pairs of parallel sides. Geometry

The opposite sides of a parallelogram are parallel and are of equal length.
The opposite angles of a parallelogram are also equal. That is, ∠ A = ∠ C; ∠ B = ∠ D

Now, let us see some of the most well known parallelograms.

Rhombus

It is a Parallelogram that has all four sides of equal length. Geometry

It is also called as equilateral quadrilateral, because it has four equal sides (just as we call a triangle with three equal sides equilateral triangle).

So, Perimeter of a Rhombus = 4 × Side

Parallelogram circumscribing a circle is a Rhombus. Geometry

Rectangle

It is a Parallelogram that has two diagonals of equal length, and all its four angles are right angles too (just like a square). However, only its opposite sides are equal.

So, its adjacent sides make an angle of 90° with each other. Geometry

AB = DC; BC = AD
AC = BD
∠ A = ∠ B = ∠ C = ∠ D = 90°

Perimeter of rectangle = 2 (length + breadth) = 2 (l + b)

If all the four sides of a rectangle are equal, then it is called a Square. So, square is a special case of rectangle.

Parallelogram inscribed in a circle is a Rectangle. Geometry

Square

It is a Parallelogram that not only has all four sides of equal length (just like a rhombus), but all its angles are right angles too (just like a rectangle).
Geometry

So, Perimeter of a Square = 4 × Side

Square - It is a Rectangle that has four sides of equal length.

That is, all of its four angles are 90° each, and all of its four sides are equal too.

So, in a way, square is a special case of rectangle.

In fact, Square is a Rectangle, as well as a Rhombus. However, vice-versa need not be true, i.e. a Rectangle or a Rhombus need not be a Square.

Properties of Parallelogram

These are the properties that are true for any kind of parallelogram.

Property 1: Angles

In a parallelogram, sum of any two consecutive angles is always supplementary. Geometry

In the above figure, ∠A + ∠B = ∠B + ∠C = ∠C + ∠D = ∠D + ∠A = 180°

In case of Rhombus: Pair of opposite angles are equal.

In case of Rectangle and Square: All angles are equal, as all angles are 90°.

Property 2: Diagonals

Property 2a

In a parallelogram, diagonals always bisect each other. Geometry

In the above figure, AO = OC; DO = OB

Property 2b

Each diagonal of a parallelogram bisects that parallelogram into two congruent triangles. Geometry

In the above figure, ∆ABD ≅ ∆CDB

Property 2c

Sum of squares of the sides of a parallelogram = Sum of the squares of its diagonals. Geometry

In the above figure, $AC^2 + BD^2 = AB^2 + BC^2 + CD^2 + DA^2$

However, in a parallelogram the opposite sides are equal. So, AB = CD, and BC = DA
So, $AC^2 + BD^2 = 2 (AB^2 + BC^2)$

In case of Rhombus:

Diagonals are not equal in length.
Diagonals bisect each other perpendicularly.
Diagonals are angle bisectors.

Sum of the square of the diagonals = Four times the square of side, i.e. $d_1^2 + d_2^2 = 4 a^2$

In case of Square:

Diagonals are equal in length (unlike Rhombus).
Diagonals bisect each other perpendicularly.
Diagonals are angle bisectors.

Diagonal of a square = $\sqrt{2}$ Side, i.e. d = $\sqrt{2}$ a

In case of Rectangle:

Diagonals are equal in length (like Square, unlike Rhombus).
Diagonals bisect each other (like Square and Rhombus), but not perpendicularly. (unlike Square and Rhombus)
Diagonals are not angle bisectors. (unlike Square and Rhombus)