#### Thank you for registering.

One of our academic counsellors will contact you within 1 working day.

Click to Chat

1800-5470-145

+91 7353221155

CART 0

• 0
MY CART (5)

Use Coupon: CART20 and get 20% off on all online Study Material

ITEM
DETAILS
MRP
DISCOUNT
FINAL PRICE
Total Price: Rs.

There are no items in this cart.
Continue Shopping

# Revision Notes on Understanding Quadrilaterals

### Plane Surface

A flat surface like paper is a plane surface.

## Plane Curve

When we get a curve by joining the number of points without lifting the pencil is a plane curve.

It could be an open or closed curve.

## Polygons

The simple closed curves which are made up of line segments only are called the Polygons.

### Classification of Polygons

Polygons can be classified by the number of sides or vertices they have.

 Number of sides Name of Polygon Figure 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon n n-gon

## Diagonals

Any line segment which connects the two non-consecutive vertices of a polygon is called Diagonal.

## Interior and Exterior of a Closed Curve

The blue shaded part represents the interior and exterior of the closed curve.

## Convex and Concave Polygons

The polygons which have all the diagonals inside the figure are known as a Convex Polygon.

The polygons which have some of its diagonals outside the figure also are known as a Concave Polygon.

## Regular and Irregular Polygons

Polygons which are equiangular and equilateral are called Regular Polygons i.e. a polygon is regular if-

• It’s all sides are equal.

• It’s all angles are equal.

Hence square is a regular polygon but a rectangle is not as its angles are equal but sides are not equal.

## Angle Sum Property

The sum of all the interior angles of a polygon remains the same according to the number of sides regardless of the shape of the polygon.

The sum of interior angles of a polygon is-

(n - 2) × 180°

Where n = number of sides of the polygon

Example

 Polygon Number of Sides Sum of Interior Angles Triangle 3 (3 – 2) × 180° = 180° Quadrilateral 4 (4 – 2)* × 180° = 360° n-gon n (n – 2) × 180°

Remark: This property is applicable to both convex and concave polygon.

### Sum of the Measures of the Exterior Angles of a Polygon

The sum of the exterior angles of any polygon will be 360°.

This is used to find the number of sides in a regular polygon.

This is applicable to irregular polygon also. The sum will remain the same whether it is a regular or irregular, small or large polygon.

Sum of all the exterior angles in the above irregular pentagon is

102° + 81° + 63° +90° + 24° = 360°

Any closed polygon with four sides, four angles and four vertices are known as Quadrilateral. It could be a regular or irregular polygon.

### Angle sum property of a Quadrilateral

• Sum of all the interior angles of a Quadrilateral = 360°

• Sum of all the exterior angles of a Quadrilateral = 360°

There are different types of the quadrilateral on the basis of their nature of sides and their angle.

1. Trapezium

If a quadrilateral has one pair of parallel sides then it is a Trapezium.

Remark: If the non-parallel sides of a trapezium are equal then it is called Isosceles Trapezium.

2. Kite

If the two pairs of adjacent sides are equal in a quadrilateral then it is called a Kite.

Here AB = BC and AD = CD

Properties of a kite

• The two diagonals are perpendicular to each other.

• One of the diagonal bisects the other one.

• ∠A = ∠C but ∠B ≠∠D

3. Parallelogram

If the two pairs of opposite sides are parallel in a quadrilateral then it is called a Parallelogram.

Here, AB ∥ DC and BC ∥ AD, hence ABCD is a parallelogram.

## Elements of a Parallelogram

Some terms related to a parallelogram ABCD

1. Opposite Sides – Pair of opposite sides are

AB and DC,

2. Opposite Angles – Pair of opposite angles are

∠ A and ∠C

∠B and ∠D

AB and BC

BC and DC

∠A and ∠B

∠B and ∠C

∠C and ∠D

∠A and ∠D

### Properties of a Parallelogram

1. The opposite sides of a parallelogram will always be equal.

Here, AB = DC and AD = BC.

2. The opposite angles of a parallelogram will always be of equal measure.

As in the above figure, ∠A = ∠C and ∠D = ∠B.

3. The two diagonals of a parallelogram bisect each other.

Here in ABCD, AC and BD bisect each other at point E. So that AE = EC and DE= EB.

4. The pair of adjacent angles in a parallelogram will always be a supplementary angle.

Example

If the opposite angles of a parallelogram are (3x + 5) ° and (61– x) °, then calculate all the four angles of the parallelogram.

Solution

As we know that the opposite angles are equal in a parallelogram so

(3x + 5)° = (61 – x) °

3x + x = 61– 5

4x = 56

x = 14°

Now substitute the value of x in the given angles.

(3x + 5)° = 3(14) + 5

= 42 + 5 = 47°

(61 – x)° = 61 – 14

= 47°

Both the angles are 47° as the opposite angles are equal.

Now to find the other angles let one of the adjacent angles to the above angle is z.

47° + z = 180° (adjacent angles are supplementary angles).

Z = 180° - 47°

= 133°

The fourth angle will also be 133° as the opposite angles are equal.

Hence the four angles of the given parallelogram are 47°, 47°, 133° and 133°.

## Some Special Parallelograms

1. Rhombus

If a parallelogram has all the four sides equal then it is called a Rhombus.

Here, AB = BC = CD = AD.

Properties of a Rhombus

• A rhombus has all the properties of a parallelogram and a kite.

• The special property is that its diagonals are perpendicular bisector to each other.

• Hence, AE = EC and DE = EB.

• It makes an angle of 90° at the point of intersection of the diagonals.

2. Rectangle

A parallelogram with equal angles and equal opposite sides is called Rectangle.

Here, AB = DC and AD = BC.

Properties of a Rectangle

• Opposite sides are parallel and equal.

• Opposite angles are equal.

• Adjacent angles make a pair of supplementary angles.

• Diagonals are of equal length.

• Diagonals bisect each other.

• All the four angles are of 90°.

3. Square

A rectangle with all the four equal sides is called a Square.

Here, AB = BC = CD = AD.

Properties of a Square

• All four sides are equal.

• Opposite sides are parallel.

• Diagonals are of equal length.

• Diagonals are perpendicular bisector to each other.

• All the four angles are of 90°.