When we think of a room, its walls and floors, a news paper, a note book, the computer screen, the surfaces of a rectangular box etc, we see flat surfaces which are rectangular. The rectangular surface is the one which has four sided and the sides are straight. So we can see that the rectangular surfaces are those which are bounded by four sides. All four sided figures are not rectangles. The four sided figures called quadrilaterals are classified into different types according to the sides and the angles. In this section let us see the definition of quadrilaterals and the other types of quadrilaterals and their properties.

What is a Quadrilateral? Quadrilateral is a four sided polygon. It is a closed figure bounded by four sides. Example: Floor of a room, walls of a room, ceiling of a room, computer screen, The surface of a note book.
Quadrilateral

Types of Quadrilaterals:

The quadrilaterals are classified into different names according to the relationship of their sides and the angles.
The different types of Quadrilaterals are, Parallelogram, rhombus, rectangle, square, trapezoid, concave and convex quadrilaterals.
Types of Quadrilaterals
1. Parallelogram: It is a quadrilateral whose pair of opposite sides are parallel.

2. Rhombus: It is a parallelogram whose sides are of congruent.

3. Rectangle: It is a parallelogram whose interior angles measure 90o .

4. Square: It is a parallelogram whose sides are congruent and the interior angles measure 90o .

5. Trapezoid: It is a quadrilateral whose one pair of opposite sides are parallel.

6. Kite: It is a quadrilateral which has two distinct pairs of adjacent sides are congruent.
Type of Quadrilaterals
Regular Quadrilateral: Regular Quadrilaterals are those in which the sides are of equal length and interior angles are equal. Square is a regular quadrilateral whose sides are of equal measure and the interior angles measure 90o .

Irregular Quadrilateral: The quadrilaterals which are not regular, are called irregular quadrilaterals. In an irregular quadrilateral, all sides are not equal and the interior angles are not of same measure.
Quadrilaterals Types
Concave Quadrilateral: Concave quadrilaterals, are those in which one of its interior angle measure more than 180o (reflex angle). The above diagram shows a concave quadrilateral.

Convex Quadrilateral: A quadrilateral whose interior angles measure less than 1800 ( acute or obtuse ) is called a convex quadrilateral.

Square, Rectangle, Parallelogram, Rhombus, Trapezoid are all convex quadrilateral.

Properties of Quadrilaterals:


1. Properties of a Parallelograms:
a. Opposite sides are parallel.
b. Opposite sides are congruent.
c. Opposite angles are congruent.
d. Pair of consecutive angles are supplementary.
e. Diagonals bisect each other.

2. Properties of a Rhombus:
a. Opposite sides are parallel.
b. All the four sides congruent.
c. Opposite angles are congruent.
d. Pair of consecutive angles are supplementary.
e. Diagonals bisect each other.

3. Properties of a Rectangle:
a. Opposite sides are parallel.
b. Opposite sides are congruent.
c. Each of interior angle measure 90o .
d. Diagonals are congruent.
e. Diagonals bisect each other.

4. Properties of a Square:
a. Opposite sides are parallel.
b. All the four sides are congruent.
c. Each of interior angle measure 90o.
d. Diagonals are congruent.
e. Diagonals bisect each other.

5. Properties of a Trapezoid:
a. It has one pair of opposite sides parallel.
b. The interior angles on the same side of the non-parallel lines are supplementary.

6. Properties of Isosceles Trapezoid:
a. It has one pair of opposite sides parallel.
b. The non parallel sides are congruent.
c. The lower base angles are congruent.
d. The upper base angles are congruent.
e. Diagonals are congruent.
f. Pair of a lower base angle and an upper base angle is supplementary.

Similar Quadrilaterals: The pair of quadrilaterals are said to be similar if their corresponding interior angles are equal and the corresponding sides are proportional. The following diagram shows pairs of quadrilaterals which are similar.

Cyclic Quadrilaterals: Quadrilaterals whose vertices lie on the circumference of a circle are called cyclic quadrilaterals.
The following diagram shows the cyclic quadrilaterals.

The Properties of Cyclic Quadrilaterals:
1. Opposite angles of a cyclic quadrilateral is equal to 180o .

2. Exterior angles of a cyclic Quadrilateral is equal to the interior opposite angle.
Sum of all the angles of a quadrilateral is equal to 360o

Solved Example

Question: The interior angles of quadrilateral are in the ratio 3 : 5 : 7 : 9, Find the measure of each angle of the quadrilateral.

Solution:
 
We are given that the angles are in the ratio 3 : 5 : 7 : 9
Let us assume that the measure of each angle is 3x, 5x, 7x and 9x.
We know that the sum of all the four angles is equal to 360o .
Therefore, by adding the angles, we get 3x + 5x + 7x + 9x = 360
=> 24 x = 360
=> x = $\frac{360}{24}$

= 15
Measure of each angle will be, 3x = 3 (15) = 45o
5x = 5 (15) = 75o
7x = 7 (15) = 105o
9x = 9 (15) = 135o

Therefore, the measure of each interior angles are, 45o , 75o , 105o , 135o .
 

Area of Quadrilateral: Area of a quadrilateral is the sum of the areas of the two triangles formed by one of its diagonal.

If the perpendicular drawn to one of the diagonals from the opposite vertices are given by h1 and h2 .
Then the area of each triangle = $\frac{1}{2}$ x diagonal x h1 and $\frac{1}{2}$ x diagonal x h2

Therefore, sum of the areas of the two triangles = $\frac{1}{2}$ x diagonal x h1 + $\frac{1}{2}$ x diagonal x h2

= $\frac{1}{2}$ x diagonal x (h1 + h2)

Areas of different types of quadrilaterals:

1. Area of a parallelogram = base x Height = bh
2. Area of a rectangle = length x width = l w
3. Area of a square = side x side = s2

4. Area of a rhombus = $\frac{1}{2}$ Diagonal 1 x Diagonal 2

5. Area of a trapezoid = $\frac{1}{2}$ x Height x (sum of the parallel sides)

= $\frac{1}{2}$ x h x ( a + b) , where h is the perpendicular distances between the parallel sides a and b.

Perimeter of a quadrilateral: Perimeter of a quadrilateral is the sum of all the sides.

Perimeter of different types of quadrilaterals:
1. Perimeter of a parallelogram = 2 (sum of the two adjacent sides)
2. Perimeter of a rectangle = 2 (length + width)
3. Perimeter of a square = 4 x side = 4 s
4. Perimeter of a rhombus = 4 x side = 4 s
5. Perimeter of a trapezoid = sum of the parallel sides + sum of non-parallel sides.

Prove that the opposite angles of a parallelogram are congruent.

Quadrilateral Proofs
Given : In the above diagram, ABCD is a parallelogram.

To Prove : $\angle A$ = $\angle C$ and $\angle B$ = $\angle D$

Construction: Join AC.

Sl. No
Statement Reason
1. In $\Delta ABC$ and $\Delta ADC$,
$\angle BAC$ = $\angle ACD$
Alternate Angles are equal
2. $\angle ACB$ = $\angle CAD$ Alternate Angles are equal
3. AC = AC Common Side
4. $\Delta ABC$ $\cong $ $\Delta ADC$ SAS congruence
5. $\angle B$ = $\angle D$ Congruent parts of congruent triangles are equal
6. Similarly, $\angle A$ = $\angle C$

Therefore, we have proved that in a parallelogram, the opposite angles are equal.

Solved Examples

Question 1: In a parallelogram ABCD, if $\angle A$ = (2x + 36)o and $\angle C$ = (3x - 5) o
a. Find the value of x.
b. Measure of each angle of the parallelogram ABCD.
Quadrilateral Problems
Solution:
 
We have In a parallelogram ABCD, if $\angle A$ = (2x + 36)o and $\angle C$ = (3x - 5) o

Since the opposite angles of a parallelogram are equal,
$\angle A$ = $\angle C$
=> 2x + 36 = 3x - 5
=> 2x - 3x = -5 - 36
=> - x = - 41
=> x = 41o
Substituting, $\angle A$ = (2x + 36)o
= 2 (41) + 36
= 118o
Since the adjacent angles of a parallelogram are supplementary,
$\angle A$ + $\angle B$ = 180
=> 118 + $\angle B$ = 180
=> $\angle B$ = 180 - 118
= 62o
since $\angle B$ = $\angle D$
we get $\angle D$ = 62o
Therefore, the angles of the quadrilateral are , $\angle A$ = 118o , $\angle B$ = 62o , $\angle C$ = 118o , $\angle D$ = 62o
 

Question 2: In the figure given below, ABCD is a square, and CDE is an equilateral triangle.
Find, $\angle AED$.
Quadrilateral Solved Examples
Solution:
 
Given : In the given figure ABCD is a square and $\Delta CED$ is an equilateral triangle.
To find :$\angle AED$
Proof :

1.
Statement
Reason
2. AB = BC = CD = DA ABCD is a square
3. DE = EC = CD $\Delta DEF$ is equilateral
4. $\angle ADC$ = 90o Interior angle of the square
5. $\angle EDC$ = 60o Interior angle of the equilateral angle
6. $\angle ADE$ = 90 - 60 = 30o (4) - (5)
7. In $\Delta ADE$ , AD = DE From (1) and (2)
8. $\angle DAE$ = $\angle DEA$ = x (say) Angles opposite to the equal sides of an isosceles
triangle are equal
9. $\Delta ADE$,
$\angle ADE$ + $\angle DAE$ + $\angle DEA$ = 180o

Angle sum property of a triangle
10. (i. e) 30 + x + x = 180
=> 30 + 2x = 180
=> 2x = 180 - 30
= 150
=> x = $\frac{150}{2}$
= 75o
From (6) and (8)

Therefore, $\angle AED$ = 75o