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.

## Properties of Quadrilateral

### what are **Quadrilaterals**?

Polygon having four sides are called quadrilaterals

**Properties of a quadrilateral **

#### 1.Side of Quadrilateral

There are four sides and quadrilateral

** Four sides –** AB, BC, CD and DA

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

#### 2.Vertices of Quadrilateral

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

#### 3. Adjacent sides and angles of Quadrilateral

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

Pair of Adjacent sides in the above quadrilateral are –

AB and BC

BC and CD

CD and DA

DA and AB

**Pair of opposite sides in the above quadrilateral are –**

AB and CD

BC and AD

**Pair of Adjacent angles in the above quadrilateral are** –

∠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

#### 4. Diagonals of Quadrilateral

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 360^{o}

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

^{o}

127** ^{o }+ **53

**127**

^{o}+**53**

^{o }+**360**

^{o}=

^{o}## Types of Quadrilateral

Here are the list of category of quadrilateral which will will help you understand about the geometry in detail

### Category of Quadrilateral

#### 1.Regular Quadrilateral

A **regular quadrilateral** is a quadrilateral with all sides having equal length.

Ex – Only Rhombus and Square are examples of regular quadrilateral

2. Irregular 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

#### 3. Convex 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

**4. Concave Quadrilateral**

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

#### 5. **Intersecting Quadrilateral**

If pair of non-adjacent sides intersect each other then such quadrilaterals are called intersecting quadrilaterals or self-intersecting or crossed quadrilaterals

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

## Types of Quadrilateral

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 90
(i.e., right angle)^{o} - 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 90^{o}

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 180
^{o} - 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 90^{o}

Here, the opposite angles are equal

&

Also, the adjacent angles when added (127** ^{o }+ **53

**) are equal to 180**

^{o}

^{o}**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 90^{o}

OA = OB = OC = OD

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

**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 180
^{o}

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 = 70** ^{o }**and ∠B = ∠D = 110

^{o}Adjacent angles are supplementary (∠A + ∠B = 110

**70**

^{o}+**180**

^{o}=

^{o})**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 (180
)^{o} - 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 = 110** ^{o} + **70

**180**

^{o}=**and ∠B + ∠C = 110**

^{o}**70**

^{o}+**180**

^{o}=

^{o})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 = 125^{o}

**Table showing the properties of different types of quadrilateral**

**Quadrilateral Formulas**

**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