# Quadrilateral: Definition, Types and Properties

In this post will study the definition and properties of quadrilateral. The concept is explained in very simple manner so that no room of doubt is left among the students. In case if the concept is left unclear, feel free to ask in the comment section.

Polygon having four sides are called quadrilaterals

There are four sides and quadrilateral

Four sides – AB, BC, CD and DA

Four angles – ∠ABC, ∠BCD, ∠CDA, and ∠DAB

The end points of the same side are adjacent vertices

Four vertices – Points A, B, C and D

Side AB have two end points A and B. that’s why they are adjacent vertices

Any two sides with a common end point are adjacent sides

Side AB and BC are adjacent side as they have common end point B. Similarly, BC & CD, CD & DA, DA & AB are also adjacent sides

AB and BC

BC and CD

CD and DA

DA and AB

Pair of opposite sides in the above quadrilateral are –

AB and CD

∠A and ∠B

∠B and ∠C

∠C and ∠D

∠D and ∠A

Pair of opposite angles in the above figure are

∠A and ∠C

∠B and ∠D

All quadrilaterals have 2 diagonals. (The join of any two non-adjacent vertices in a quadrilateral is a diagonal

#### 5. Interior Angles of Quadrilateral

The sum of all the interior angles of a quadrilateral is equal to 360o

#### Here, the sum of all the interior angles is equal to 360o

127o + 53o + 127o + 53o = 360o

Ex – Only Rhombus and Square are examples of regular quadrilateral

An irregular quadrilateral is a quadrilateral with sides that are not all equal in length.
Ex – Rectangle, parallelogram, etc. are examples of irregular quadrilateral

If both the diagonals of a quadrilateral are completely contained within the figure, then it is called convex quadrilateral

Here both diagonals AC and BD are contained within the figure

If at least one of the diagonals of a quadrilateral lie either partly or completely outside of the figure, then it is called concave quadrilateral

The diagonal AC lies outside the figure

Here, non-adjacent sides (AC & BD) intersect each other

Some of the types of quadrilateral are mentioned below
a. Square
b. Rhombus
c. Rectangle
d. Parallelogram
e. Kite
f. Trapezium

### Properties of Square

• All the sides of the square are of equal measure
• Opposite sides of square are parallel to each other
• All the interior angles of a square are at 90o (i.e., right angle)
• The diagonals of a square perpendicular bisector of each other

Here, ABCD is a square with all its sides (AB, BC, CD and DA) equal
Each interior angle is equal to 90o
Opposite sides i.e., (AB & CD) and (BC & AD) are parallel to each other

OA = OB = OC = OD

Here, ABCD is a square with diagonals AC and BD which are perpendicular to each other and also bisect (divide into two equal half) each other

### Properties of Rhombus

• All the four sides of a rhombus are of the same length
• Opposite sides of the rhombus are parallel to each other
• Opposite angles of rhombus are equal
• The sum of any two adjacent angles of a rhombus is equal to 180o
• The diagonals are perpendicular bisector of each other.

OB = OD
OA = OC

Here, ABCD is a rhombus with all its sides (AB, BC, CD and DA) equal
Opposite sides i.e., (AB & CD) and (BC & AD) are parallel to each other
Diagonals AC and BD bisect each other and make an angle of 90o

Here, the opposite angles are equal
&
Also, the adjacent angles when added (127o + 53o) are equal to 180o

### Properties of Rectangle

• The opposite sides of a rectangle are of equal length
• The opposite sides are parallel to each other
• All the interior angles of a rectangle are at 90 degrees.

The diagonals of a rectangle bisect each other

Here, ABCD is a rectangle whose opposite sides (AB = CD = 8 cm and BC = AD = 5 cm) are equal
Opposite sides are parallel i.e., AB || CD and BC || AD
Each interior angle is equal to 90o

OA = OB = OC = OD

Diagonals AC and BD bisect each other at O but they are not making 90o

### Properties of Parallelogram

• The opposite side of the parallelogram are of the same length
• The opposite sides are parallel to each other
• The diagonals of a parallelogram bisect each other
• The opposite angles are of equal measure
• The sum of two adjacent angles of a parallelogram is equal to 180o

OB = OD = 4 cm
OA = OC = 5 cm

Here, ABCD is a parallelogram whose opposite sides (AB = CD = 8 cm and BC = AD = 6 cm) are equal
Opposite sides are parallel i.e., AB || CD and BC || AD
The diagonals bisect each other (divide into 2 equal halves)

Opposite angles are of equal measure (∠A = ∠C = 70o and ∠B = ∠D = 110o
Adjacent angles are supplementary (∠A + ∠B = 110o + 70o = 180o)

### Properties of Trapezium

• Only one pair of the opposite side of a trapezium is parallel to each other
• The two adjacent sides of a trapezium are supplementary (180o)
• Non-parallel sides are equal in length
• The diagonals of a trapezium bisect each other in the same ratio

Here, ABCD is a trapezium whose only one pair of opposite side is parallel i.e., AB || CD
The two adjacent sides of trapezium are supplementary (i.e., ∠A + ∠D = 110o + 70o = 180o and ∠B + ∠C = 110o + 70o = 180o)

Non-parallel sides are equal in length (AD = BC = 6 cm)
The diagonals bisect each other in same ratio, here, AP:PC = BP:PD = 6:11

### Properties of Kite

• The pair of adjacent sides of a kite are of the same length
• The largest diagonal of a kite bisects the smallest diagonal
• Only one pair of opposite angles are equal

Here, ABCD is a Kite whose pair of adjacent sides are of equal length (AB = AD = 4 cm and BC = CD = 8 cm)
Diagonal AC (longest) bisect BD (smallest)
Only one pair of opposite angles are equal i.e., ∠D = ∠B = 125o

Table showing the properties of different types of quadrilateral

### Square

Perimeter = 4 x length of each side = 4 x s

Area = side x side = s x s

### Rectangle

Perimeter = 2 x (length of rectangle + breadth of rectangle)

= 2 x (L + B)

Area = Length x Breadth = L x B

### Rhombus

Perimeter = 4 x length of each side = 4 x s

Area = side x side = s x s

### Parallelogram

Perimeter = 2 x (length of rectangle + breadth of rectangle)

= 2 x (L + B)

Area = Length x Breadth = L x B

### Trapezium

Perimeter = sum of all sides = a + b + c + d

Area = Length x Breadth = a + b x h

### Kite

Perimeter = 2 x (a + b)

Area = side x side = p x q