Parallelogram Basics

## What is a Parallelogram?

Parallelogram is a quadrilateral with two pairs of parallel sides.

- 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.

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

### 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.

AB = DC; BC = AD

AC = BD

∠ A = ∠ B = ∠ C = ∠ D = 90°

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

### 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).

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.

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.

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

#### Property 2b

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

In the above figure, ∆ABD ≅ ∆CDB

#### Property 2c

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

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)

Diagonal of a rectangle = $\sqrt{l^2 + b^2}$

### Property 3: Angle bisectors of Parallelogram

#### Property 3a

In a parallelogram, bisectors of any two consecutive angles make an angle of 90°.

In the above figure, ∠AOB = 90°

#### Property 3b

Bisectors of the four angles of a parallelogram form a rectangle.

In the above figure, PQRS is a rectangle.

However, there are a few properties that are true only for some specific kinds of parallelograms. Let’s see these too.

### Property 4: Mid-point of sides

#### In case of Rhombus

Line segments joining midpoints of sides of a rhombus form a rectangle.

In the above figure, PQRS is a rectangle.

#### In case of Square

Line segments joining midpoints of sides of a square form a square.

In the above figure, PQRS is a square.

#### In case of Rectangle

Line segments joining midpoints of sides of a rectangle form a rhombus.

In the above figure, PQRS is a rhombus.

### Property 5: In case of Rectangle

If P is some point inside of a rectangle, then:

$PA^2 + PC^2 = PB^2 + PD^2$