**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**(b) Breadth**

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

AD = BC

**(02) Opposite sides are parallel**

In the above figure;

AB is parallel to CD

AD is parallel to BC

**(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**

∠ADC = ∠CBA

**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)

Adding eq(1) and eq (2)

∠1 + ∠3 = ∠2 + ∠4

∠BCA = ∠ADC

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

**Frequently Asked Questions : Parallelogram**

**(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**