[Maths Class Notes] on Parallelogram, Trapezium, and Kite Pdf for Exam

A parallelogram is a quadrilateral with two of its sides parallel. The opposite sides and angles of a parallelogram are equal. The area of a parallelogram relies on its base and height.

In the above figure, you can see,

AB//CD, AD//BC

Also, AB = CD and AD = BC

And, ∠A =∠C, ∠B =∠D

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Properties of a Parallelogram

Here are the different properties of parallelogram:

• The opposite sides of a parallelogram are congruent

• The opposite angles of a parallelogram are congruent

• The consecutive angles of a parallelogram are supplementary

• The diagonal of a parallelogram always bisect each other

• Each diagonal of a parallelogram bisect it into two congruent triangles

If any of the angles of a parallelogram is a right angle, then its other angles will also be a right angle.

Types of a Parallelogram

The three different types of a parallelogram are:

Trapezium 

The trapezium is a type of quadrilateral with two of its sides parallel. The parallel sides of a trapezium are called bases whereas non-parallel sides of a trapezium are called legs. The trapezium is also known as a trapezoid. Sometimes, the parallelogram is also considered as a trapezoid with two of its sides parallel.

In the above figure, we can see sides AB and CD are parallel to each other whereas sides BC and AD are non-parallel. The h is the distance between the two parallel sides which represent the height of the trapezium.

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Properties of a Trapezium

Here are the different properties of a trapezium:

• One pair of opposite sides are parallel in trapezium

• The diagonals of the trapezium intersect each other

• The sides of a trapezium that are not parallel are not equal except in isosceles trapezium

• The sum of the interior sides of a trapezium is equal to 360 degrees i.e., ∠A +  ∠B +∠C +∠D = 360°

• The sum of two adjacent angles is equal to 180°. It implies that two adjacent angles are supplementary.

• The legs or non parallel sides of an isosceles trapezium are congruent.

Types of Trapezium

The trapezium is of three different types namely:

• Isosceles Trapezium- The legs or non parallel sides of an isosceles trapezium are equal in length.

• Scalene Trapezium- All the sides and angles of a scalene trapezium are of different measures.

• Right trapezium- A right trapezium includes at least two right angles.

Kite Definition 

A kite is a quadrilateral with two pairs of adjacent and congruent (equal-length) sides. It implies that kite is

• A polygon

• A closed shape

• A plane figure

What are the Properties of a Kite?

Here are some important properties of a kite:

• A kite is symmetrical in terms of its angles.

• The two diagonals of a kite bisect each other at 90 degrees.

• The main diagonal of a kite bisects the other diagonal.

• The smaller diagonal of a kite divides it into two isosceles triangles.

• The angles of a kite are equal whereas the unequal sides of a kite meet.

• The kite can be seen as a pair of congruent triangles with a common base.

Solved Examples

1. Find the perimeter of kite whose sides are 21cm and 15cm

Solution : 

Given

 a= 21cm

 b= 15cm

Perimeter of the kite= 2(a+b)]

Perimeter of kite 2(21+15)

Perimeter of kite = 72 cm

2. Find the area of a parallelogram whose base is 5 cm and height is 7cm.

Solution- Given, Base = 5cm and Height = 7 cm

Area= Base * Height

Area= 5 * 7

Area = 35 sq. cm

Hence, the area of a parallelogram is 35 sq cm.

3. Find the perimeter of a trapezium whose sides are 6cm ,7cm, 8cm, and 9 cm

Solution: Perimeter of trapezium= sum of all its sides

Perimeter = 6+7+8+9

Perimeter = 30cm

Hence, the perimeter of the trapezium is 30cm.

Did You know?

The word “parallelogram” has its origin in Latin roots; in the 16th century, it was formed from the Latin words “Parallelos” which literally means parallel, and “gramme” which means line.

Parallelepiped is a 3D parallelogram and is formed when you shear the opposite sides of a square or a rectangle. 

Orion Nebula, a group of stars in the sky, is also known as a trapezium. In fact, there’s a bone in our wrist which is also called a trapezium. 

It is said that Chinese philosophers Mozi and Lu Ban were the ones to come up with the shape kite in the 5th century BC. 

Quiz Time

1. Which of the following quadrilaterals is a regular quadrilateral?

1. Rectangle

2. Square

3. Rhombus

4. None of these      

2. In an isosceles parallelogram, we have

1. Pair of parallel sides equal

2. Pair off non-parallel sides equal

3. Pair of non-paralle
   l sides are perpendicular

4. None of these 

3. What do we call parallel sides of  the trapezium

1. Edges of trapezium

2. Angles of trapezium

3. Legs of trapezium

4. Bases of trapezium

4. How many pairs of equal opposite angles

1. 0

2. 1

3. 2

4. 3

5. 4