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
(Image will be uploaded soon)
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:
Square
Rectangle
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.
(Image will be uploaded soon)
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
Which of the following quadrilaterals is a regular quadrilateral?
Rectangle
Square
Rhombus
None of these
In an isosceles parallelogram, we have
Pair of parallel sides equal
Pair off non-parallel sides equal
Pair of non-parallel sides are perpendicular
None of these
What do we call parallel sides of the trapezium
Edges of trapezium
Angles of trapezium
Legs of trapezium
Bases of trapezium
How many pairs of equal opposite angles
0
1
2
3
4
FAQs on Parallelogram, Trapezium, and Kite
1. What are the important formulas of Parallelogram, Kite, and Trapezium which are 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.
The 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 on 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.
The 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
You can learn more about Parallelogram, Trapezium, and Kite in Vedantu. You will get different study materials on the Vedantu website and app which will help you to understand this topic better.
2. State some of the applications of a parallelogram and a trapezium.
The applications of each of the shapes are as follows:
1. Parallelogram
The unique properties of this shape have led to its extensive use in various industries, especially when it comes to transferring mechanical motion from one place to another.
Other uses include the shape being utilised in the structure of shed roofs and gable roofs, tiles, solar panels, the steps on a staircase, and a striped pole.
2. Trapezium
This shape is present in our daily lives in various forms including a drinking glass, a flowerpot, a ring, a bucket, a shovel, a wheelbarrow, sheds, boats, and even the locks we use for our house.
3. How do a kite and an isosceles trapezoid have dual properties?
A kite is a quadrilateral that tends to have an axis of symmetry along with one of its diagonals. Any quadrilateral that is non-self-crossing and has an axis of symmetry, is either a kite or an isosceles trapezoid.
The polar figure present in a kite is an isosceles trapezoid and vice versa, rendering both the shapes to be dual. The comparison of their dualities is as follows:
Isosceles trapezoid | Kite |
It has two pairs of equal adjacent angles. | Pairs of adjacent sides are equal for a Kite. |
One pair of opposite angles are equal for an Isosceles trapezoid. | It has one pair of opposite angles that are equal. |
It is a circumscribed circle. | It is an inscribed circle. |
It has an axis of symmetry through a pair of its opposite sides. | It has an axis of symmetry through a pair of opposite angles. |
4. What is an antiparallelogram and what are its uses?
In geometry, a type of self-crossing quadrilateral is considered to be an antiparallelogram. It has two opposite pairs of sides that are equal in length, just like a parallelogram, however, these aren’t in general parallel, instead, the sides that are in the longer pair end up crossing each other in a sort of a scissor mechanism. An antiparallelogram is also commonly known as a crossed parallelogram or a contraparallelogram. Each and every antiparallelogram tends to have an axis of symmetry through its crossing point.
Some of the uses of an antiparallelogram are as follows:
Various non-convex uniform polyhedra tend to have an antiparallelogram as their vertex figure.
The construction of ellipses from the motion of an antiparallelogram can be utilized in the design and mechanism of elliptical gears which further convert uniform motion into a non-uniform motion and/or vice versa.
This particular shape is also used as a form of four-bar linkage wherein four rigid beams that are of a fixed length are likely to rotate with respect to each other at joints that are placed at the four vertices of the antiparallelogram.
5. Is a square and a rectangle considered a parallelogram?
Yes, both square and rectangle are 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 the measure of every interior angle is 90 degrees. It is also known as an equiangular quadrilateral.
Rectangle also has similar properties to parallelograms such as the opposite sides of a rectangle are parallel.