Parallelogram (Geometry) : Definition & Properties

What is Parallelogram?

Parallelogram is a quadrilateral in which opposite sides are equal & parallel and opposite angles are equal.

Keyword for Parallelogram
(a) Opposite sides are equal & parallel
(b) Opposite angles are equal

Structure for Parallelogram

Parallelogram consists of following components

(a) Length
The longer side is known as Length

The shorter side is the breadth

(c) Vertex
The point where two sides meet is called vertex
There are four vertex in parallelogram : A, B, C & D

(d) Angles
Four angles are present in parallelogram.
Opposite angles are equal
∠A = ∠C
∠D = ∠B

Properties of Parallelogram

(01) Opposite sides are equal

Given above is figure of parallelogram in which:
AB = CD

(02) Opposite sides are parallel

In the above figure;
AB is parallel to CD

(03) Opposite angles are equal

∠A = ∠C
∠D = ∠B

Let us prove the above concept

Theorem
In parallelogram opposite angles are equal

Given
Given below is parallelogram ABCD with diagonal BD
AB = CD and AD = BC

To prove

Solution
We know that in parallelogram opposite sides are parallel
So, AB is parallel to CD and AD is parallel to BC

Since,
AB || CD and BD is a transversal;
∠1 = ∠2 { Alternate Angles } – – – eq(1)

Similarly,
AD || BC and BD is a transversal;
∠3 = ∠4 { Alternate Angles } – – – eq(2)

∠1 + ∠3 = ∠2 + ∠4

Hence proved that opposite angles are equal

(04) The sum of all interior angles of parallelogram adds to 360 degree

Given above is parallelogram ABCD with diagonal BD

The diagonal divides the parallelogram into two triangles ABD and BCD

Taking ▵ ABD and using angle sum property
∠1 + ∠2 + ∠3 = 180 degree – – – -eq(1)

Now taking ▵BCD
∠4 + ∠5 + ∠6 = 180 degree – – – -eq(2)

Adding eq(1) & eq(2), we get

∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 = 180 + 180

∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 = 360 degree

Hence, sum of all interior angle of parallelogram is 360 degree

(05) In Parallelogram, consecutive angles are supplementary (180 degree)

We know that in parallelogram, opposite sides are parallel to each other (i.e AB || CD)

Considering AB & CD as parallel lines and AD as a transversal, we can say that:
∠BAD + ∠CDA = 180 degree { same side interior angles}

Hence, ∠BAD & ∠CDA are supplementary angles.

Similar is the case for below pairs
∠D + ∠C = 180 degree
∠B + ∠C = 180 degree
∠A + ∠B = 180 degree

(06) In Parallelogram, diagonals bisect each other

Here AC and BD are the two diagonals that bisect each other.
i.e. AO = OC and BO = OD

(07) Parallelogram Law

The law states that the sum of square of all sides of parallelogram is equal to sum of square of the diagonals

Given above is parallelogram of side a cm and b cm
The length of the diagonals are p & q cm

According to Parallelogram law
a^{2} +b^{2} +a^{2} +b^{2} \ =\ p^{2} +q^{2}\\\ \\ 2\ \left( \ a^{2} +b^{2}\right) \ =\ p^{2} +q^{2}

(08) Perimeter of Parallelogram

The perimeter is calculated by finding the length of total boundary

The perimeter of parallelogram is given by formula:
Perimeter = 2 x (Length + Breadth)

(09) Area of Parallelogram

There are three formulas of area calculation for parallelogram. Each formula is used as per the data provided in the question.

Formula 01
When base length & height is given

Area of parallelogram = Base x Height = b x h

Formula 02
When length of two diagonals and intersection angle is given

Given above is the parallelogram ABCD with two diagonals of length d1 & d2, intersecting at angle 𝜃

Then the area of parallelogram is given by formula:
Area = (1/2) x d1 x d2 x sin(𝜃)

Note: sin(𝜃) is a trigonometry term used in higher mathematics.
If are in lower grade, skip this formula for future learning

Formula 03
When length of two sides and angles between them is given

Given above is the parallelogram with sides a cm & b cm and angle between the side is x degree

The formula for Area of Parallelogram is:
Area = a x b x sin(x)

(10) Line of symmetry

A line which divides the figure into two equal halves is called line of symmetry.

A parallelogram has no line of symmetry

(01) How is parallelogram and square different

In Square
(a) All sides are equal and parallel
(b) All angles are 90 degree

But in Parallelogram
(a) Opposite sides are equal
(b) Angles may not be 90 degrees

(02) Are squares a form of parallelogram ?

YES !!

Above is the figure of a square.

The figure also fulfills all property of parallelogram
(a) Opposite sides equal & parallel
(b) Opposite angles equal

Hence, all squares are parallelogram

(03) Can all parallelograms be square?

NO!!

In the above parallelogram, all sides are not equal and angles are also not 90 degree

Hence, all parallelogram are not squares.
But all squares are parallelogram

Solved Questions – Parallelogram

(01) Find the area of parallelogram if base length is 3 cm and height is 5 cm

Area of Parallelogram = Base x Height
Area of Parallelogram = 3 x 5 = 15 sq cm

(02) Find the perimeter of parallelogram if the length of adjacent sides are 3 cm and 6 cm

Solution
We know that:
Perimeter of Parallelogram = 2 ( Length + Breadth)
Perimeter = 2 ( 3 + 6 ) = 2 x 9 = 18 cm

Hence, 18 cm is the answer

(03) The length of two diagonals of parallelogram are 4 cm and 5 cm. The diagonals intersect each other at 30 degree. Find the area of parallelogram
{ Sin 30 = 1/2 }

Solution
d1 = 4 cm
d2 = 5 cm
𝜃 = 30 degree

Area of Parallelogram = (1/2) x d1 x d2 x sin(𝜃)

Area of Parallelogram = (1/2) x 4 x 5 x (1/2) = 5 sq. cm

Hence 5 sq cm is the required area