Parallelogram

Parallelogram Definition

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 the parallelogram are:

  1. Square

  2. Rectangle

  3. Rhombus


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 trapezium intersect each other

  • The sides of a trapezium which 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:

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

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

  3. 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


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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


  1. 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.


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


Solution: Perimeter of trapezium= sum of all its sides


Perimeter = 6 + 7 + 8 + 9


Perimetre = 30 cm


Hence, perimeter of trapezium is 30 cm


Quiz Time -

1. Which of the following quadrilateral 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-parallel 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

FAQ (Frequently Asked Questions)

1. What are the Important Formulas of Parallelogram Kite, and Trapezium Which is Mostly Used?

Some important formulas of Trapezium, Kite, and Parallelogram are given below.


The area of a parallelogram is the area occupied by it in a two-dimensional plane. 


Area of a parallelogram = Base × Height


The perimeter of a parallelogram is the measurement is the total distance of the boundaries of a parallelogram.


Perimeter of a parallelogram = 2(a+b)

 

Here, a and b are the length of the equal sides of the parallelogram.


The area of the trapezium can be calculated by taking the average of the two bases of a  trapezium and multiplying by its altitude.


Area of a trapezium A, = h(a+b)/2

Where,

h is the height of the trapezium and a,b are the bases of a trapezium.


Perimeter of a Trapezium -

The perimeter of a trapezium is calculated by adding all its sides. Hence, the formula of the perimeter of a trapezium is given by


Perimeter of a trapezium = a + b +c+ d


Here, a,b,c and d are the sides of the trapezium.


A kite has two pairs of each side. The total distance around the outside of a kite is known as the perimeter of a kite. The perimeter of a kite is calculated by finding out the sum of the length of each pair of the equal sides of a kite.


Perimeter of a Kite = 2a +2b

Here,

a and b are the length of the first and second pair of the sides of a kite.


Area of a Kite

As we know diagonals of a kite are perpendicular. Accordingly, the area of a kite is expressed as half of the product of its diagonals which is similar to that of a rhombus. 


Area of a Kite = ½ d₁d₂

d₁ - Long pair of diagonal

d₂ - Short pair of diagonal

2. Is Square and Rectangle Considered as a Parallelogram?

The geometrical figures such as square and rectangle are both considered as parallelograms as the opposite sides of the square are parallel to each other and the diagonals of the square bisect each other. The rectangle is a special case of a parallelogram in which measures of its every interior angle is 90 degree. It is also known as an equiangular quadrilateral.


Rectangle also have similar properties of parallelograms such as the opposite sides of a rectangle are parallel to each other as parallelogram.