MATHS QUERY

## Definition

Quadrilateral is a plane closed figure formed by joining four line segments. It is a polygon with four sides. Quadrilateral has four sides, four vertices and four angles.

In the above figure, ABCD is quadrilateral with four line segments AB, BC, CD and DA. AB, BC, CD and DA are called sides of Quadrilateral ABCD. Also, A, B, C and D are called vertices of quadrilateral.

Quadrilateral ABCD has four angles viz. $$\angle DAB$$, $$\angle ABC$$, $$\angle BCD$$ and $$\angle CDA$$. We can write these angles as $$\angle A$$, $$\angle B$$, $$\angle C$$ and $$\angle D$$ respectively.

Two sides of a quadrilateral are said to be adjacent sides if they meet at a common end point. In above figure, AB and BC are adjacent sides of the quadrilateral ABCD because they have a common end point B. Also, BC and CD are adjacent sides of the quadrilateral ABCD because they have common end point C. CD and DA are adjacent sides of the quadrilateral ABCD because they have common end point D. DA and AB are also the adjacent sides of the quadrilateral ABCD because they have common end point A.

## What are Opposite Sides of a quadrilateral?

The sides of a quadrilateral are called as opposite sides if they do not have a common end point. Again, referring to the above quadrilateral ABCD, AB and CD are the opposite sides because they have no common end point. Also, AD and BC are also the opposite sides of the quadrilateral ABCD because they also have no common end point.

## What are Diagonals of a quadrilateral?

The line segment which joins the opposite vertices of a quadrilateral is called its diagonal.

In the above quadrilateral ABCD, AC and BD are two diagonals of quadrilateral.

Why?

Because AC line segment joins two opposite vertices A and C and hence AC is a diagonal of quadrilateral ABCD. Similarly, BD line segment joins two opposite vertices B and D, that makes BD the diagonal of quadrilateral ABCD.

## Which are types of quadrilateral?

As the number of sides of a quadrilateral are always fixed i.e. four, so, the types of Quadrilateral are classified based on how the length of four sides vary and how inclined the sides are.

Let’s have a look at very common types of a quadrilateral.

## Parallelogram

A quadrilateral, in which both pairs of opposite sides are parallel, is called parallelogram. Also, opposite sides of a parallelogram are equal.

Parallelogram

In above figure, ABCD is a parallelogram because opposite sides AB and CD are parallel to each other and have equal length. Also, AD and BC are parallel to each other and have equal length. Or, we can write them as:

AB = CD and AB || CD. Also, AD = BC and AD || BC

Hence, the above ABCD quadrilateral is a parallelogram.

A diagonal of parallelogram divides it into two congruent triangles and the two diagonals of a parallelogram bisect each other.

Parallelogram

In above parallelogram ABCD, the two diagonals AC and BD bisect each other at O. Therefore, OA=OC and OB=OD.

## Rectangle

A quadrilateral in which both pairs of opposite sides are parallel, equal in length and each of four angles is of $$90^0$$, is called rectangle.

Rectangle

In given figure, ABCD is a rectangle. Because, the opposite sides are parallel, i.e. AB||CD and BC||AD. Secondly, the length of opposite sides are equal, i.e. AB = CD and BC = AD. And the last, all of four angles are of $$90^0$$, i.e.

$$\angle A$$ = $$90^0$$, $$\angle B$$ = $$90^0$$, $$\angle C$$ = $$90^0$$, $$\angle D$$ = $$90^0$$

Diagonals of rectangle are always equal in length and bisect each other.

Rectangle

In above figure, AC = BD, because diagonals of rectangle are equal in length. And OA = OC and OB = OD, because diagonals of rectangle bisect each other.

## Square

A quadrilateral in which all sides are parallel, equal and each of its angle is 90 degree, is called square.

Square

In above figure, ABCD is a square because AB || CD, BC || AD and AB = BC = CD = DA. $$\angle A$$ = $$90^0$$, $$\angle B$$ = $$90^0$$, $$\angle C$$ = $$90^0$$ and $$\angle D$$ = $$90^0$$.

The diagonals of a square are equal in length and bisect each other at right angle.

Diagonals of square

In above figure, AC and BD are diagonals of the square ABCD and intersect each other at O. Both diagonals are equal in length. i.e. AC = BD.

Both diagonals bisect each other. i.e. OA = OC and OB = OD

Both diagonals intersect each other at right angle. i.e. $$\angle AOB$$ = $$\angle BOC$$ = $$\angle COD$$ = $$\angle AOD$$ = $$90^0$$

## Rhombus

A quadrilateral in which both pairs of opposite sides are parallel and all four sides are equal in length, is called rhombus. Rhombus is also called as equilateral quadrilateral.

Rhombus

In above figure, ABCD is a rhombus because AB || CD, BC || AD and AB = BC = CD = DA. The diagonals of a rhombus bisect each other at right angle.

Diagonals of rhombus

In above figure, the diagonals AC and BD bisect each other at right angle. i.e. OA = OC and OB = OD. $$\angle AOB$$ = $$\angle BOC$$ = $$\angle COD$$ = $$\angle AOD$$ = $$90^0$$

Diagonals of rhombus bisect angles

Also, in above figure, diagonals of a rhombus bisect the angles of rhombus at vertices. i.e. $$\angle 1$$ = $$\angle 2$$, $$\angle 3$$ = $$\angle 4$$, $$\angle 5$$ = $$\angle 6$$ and $$\angle 7$$ = $$\angle 8$$

## Trapezium

A quadrilateral in which only one pair of opposite sides are parallel, is called trapezium.

Trapezium

In above figure, ABCD is a trapezium because AB || CD and AD and BC are non parallel sides.

### Isosceles Trapezium

If two non parallel sides of trapezium are equal, it is called isosceles trapezium.

Isosceles Trapezium

In above isosceles trapezium ABCD, AB || CD and non parallel sides BC and AD are equal in length, i.e. BC = AD.

In isosceles trapezium, base angles are of equal in magnitude. i.e. $$\angle A$$ = $$\angle B$$ and $$\angle C$$ = $$\angle D$$

In isosceles trapezium, diagonals are equal in length. i.e. AC = BD.

## Kite

A quadrilateral in which two pairs of adjacent sides are equal, is called kite.

Kite

In above figure, ABCD is a kite because AB = BC and CD = DA.

Quadrilateral is a plane figure which is formed by joining four line segments.

### Q) What is the sum of four angles of a quadrilateral?

Sum of four angles of a quadrilateral is 3600

### Q) How many diagonals does a quadrilateral have?

There are two diagonals in a quadrilateral.

## Solved Examples

### 1) In a parallelogram, ABCD if $$\angle A$$ = 600, find the other three angles.

Solution

given $$\angle A$$ = 600

∴ $$\angle C$$ = 600

∵ $$\angle A$$ and $$\angle C$$ are opposite angles of parallelogram.

$$\angle A$$ + $$\angle B$$ = 1800

∵ adjacent angles in parallelogram are supplementary.

600 + $$\angle B$$ = 1800

$$\angle B$$ = 1800 - 600

$$\angle B$$ = 1200

∴ $$\angle D$$ = 1200

∵ $$\angle B$$ and $$\angle D$$ are opposite angles of parallelogram.

### 2) Angles of quadrilateral are 900, 800 and 1100. Find the fourth angle

Solution

Let fourth angle of quadrilateral be x.

x + 900 + 800 + 1100 = 3600 (∵ Sum of angles of quadrilateral is 3600.)

x + 2800 = 3600

x = 3600 - 2800

x = 800

### 3) In rhombus PQRS, if $$\angle P$$ = 700. Find the $$\angle Q$$, $$\angle R$$ and $$\angle S$$.

Solution

$$\angle P$$ = 700

Also, $$\angle R$$ = $$\angle P$$ (∵ opposite angles of rhombus are equal)

∴ $$\angle R$$ = 700

$$\angle P$$ + $$\angle Q$$ = 1800 (∵ Sum of adjacent angles is 1800.)

700 + $$\angle Q$$ = 1800

$$\angle Q$$ = 1800 - 700

$$\angle Q$$ = 1100

Also, $$\angle S$$ = $$\angle Q$$ (∵ opposite angles of rhombus are equal)

∴ $$\angle S$$ = 1100

## Multiple Choice Questions

a) four sides

b) five sides

c) curve

d) three sides

a) four angles

b) six angles

c) three angles

d) five angles

a) Triangle

b) Pentagon

c) Rhombus

d) Hexagon

a) 1 right angle

b) 2 right angle

c) 3 right angle

d) 4 right angle

### 5) A quadrilateral in which one pair of opposites are parallel and one pair of opposite sides are non parallel but equal in length is called as

a) Square

b) Rectangle

c) Rhombus

d) Isosceles trapezium

a) Rectangle

b) Square

c) Rhombus

d) Kite

a) Rectangle

b) Square

c) Trapezium

d) Rhombus

a) Kite

b) Rhombus

c) Square

d) Trapezium

### 9) Consecutive angles of a parallelogram are supplementary.

a) True

b) False

c) Maybe

d) None of the abve

a) Square

b) Kite

c) Rhombus

d) Parallelogram