The quadrilaterals are those which are four sided closed figures. The quadrilaterals can be classified into different types according to their properties. It is very important to know the properties of each quadrilateral as it is very much useful to design the given place or land in the required form. We have already seen the fountains at the center of a definite shaped grass land in a park. Also while fixing tiles of definite shape in a given area we should know the properties of the quadrilaterals to make our job easier. In this section let us see the types of quadrilaterals, their properties and few problems based on quadrilaterals.

## Parallelogram

Parallelogram: A parallelogram is a quadrilateral whose pair opposite sides are parallel.

Properties of a Parallelogram:

1. Opposite sides are Congruent.
In the above parallelogram //gm LANE, LA = EN and AN = LE

2. Opposite sides are parallel.
LA // EN and AN // LE

3. Pair of adjacent angles are supplementary
$\angle L$ + $\angle A$ = 180o
$\angle A$ + $\angle N$ = 180o
$\angle N$ + $\angle E$ = 180o
$\angle E$ + $\angle L$ = 180o

4. Opposite Angles are Congruent.
$\angle L$ = $\angle N$
and $\angle E$ = $\angle A$

5. Diagonals bisect each other.
AO = OE and LO = ON

6. Each diagonal divides the parallelogram into pair of congruent triangles.
$\Delta LAN$ $\cong$ $\Delta NEL$
and $\Delta ALE$ $\cong$ $\Delta ENA$

## Rectangle

Definition: Rectangle is a parallelogram whose interior angles measure right angles.

Properties of a Rectangle:

1. Opposite sides are parallel.
In the above rectangle TRUE, TR // EU and TE // RU

2. Opposite sides are Congruent.
TR = EU and TE = RU

3. Vertex angles are Congruent to 90o
$\angle T$ = $\angle R$ = $\angle U$ = $\angle E$ = 90o

4. Diagonals are congruent.
TU = ER

5. Diagonals bisect each other.
TO = OU and RO = OE

## Square

Square is a rectangle whose sides are equal.

Properties of a Square:

1. All sides are equal.
In the above square MARK, MA = AR =KR = KM

2. Each vertex angle measure 90o.
$\angle M$ = $\angle A$ = $\angle R$ =$\angle K$ = 90o

3. Diagonals are equal in measure.
KA = MR

4. Diagonals bisect each other at right angles.
MO = OR = AO = OK, MR $\perp$ AK

5. Each diagonal bisects the corresponding vertex angle.
MR is the bisector of $\angle M$ and $\angle R$.
AK is the bisector of $\angle A$ and $\angle K$.

## Rhombus

Rhombus is a parallelogram whose sides are equal.

Properties of a Rhombus:

1. Opposite sides are parallel.
DG // LO and DL // GO.

2. All sides are equal.
In the above Rhombus GOLD,
DG = GO = OL = LD

3. Opposite sides are parallel.
DL // GO and DG // LO.

4. Diagonals are perpendicular bisectors of each other.
DO $\perp$ LG

## Trapezoid

Trapezoid is a quadrilateral whose one pair of opposite sides are equal.

Isosceles Trapezoid: In the given trapezoid of the pair of non-parallel lines are congruent then the trapezoid is called an isosceles trapezoid.

Properties Isosceles Trapezoid:

1. The two non-parallel sides are of equal length.
In the above trapezoid ROME, RE = MO.

2. The base angles on each of the parallel lines are of equal measure.
$\angle R$ = $\angle O$ and $\angle E$ = $\angle M$

3. The diagonals are of equal measure.
RM = EO

4. Pair of interior angles on the same side of the non-parallel lines are supplementary.
$\angle R$ + $\angle E$ = 180o
$\angle M$ + $\angle O$ = 180o

Scalene Trapezoid: A trapezoid in which the non-parallel sides are not of equal measure is called a scalene Trapezoid.

Properties os Scalene Trapezoid:

1. One pair of opposite sides are parallel.
In the above trapezoid FOIL, FO // L I.

2. Pair of interior angles on the same side of the non-parallel lines are supplementary.
$\angle F$ + $\angle L$ = 180o
$\angle I$ + $\angle O$ = 180o

3. No sides or angles or diagonals are equal.

Right Angled Trapezoid: It is a quadrilateral which a pair of opposite sides are parallel and one pair of adjacent angles measure 90o each.

Properties of Right angled Trapezoid:
1. Pair of adjacent angles within the parallel lines measure 90o each.

2. No sides are equal.

3. The diagonals are unequal.

4. Pair of interior angles on the same side of the non-parallel lines are supplementary.
$\angle S$ + $\angle L$ = 180o
$\angle I$ + $\angle O$ = 180o

## Kite

A kite is a quadrilateral in which two pairs of adjacent sides are congruent.

In the above kite KIND, DK = DN and IK = IN.

Properties of a kite:

1. Diagonals are perpendicular to each other. DL $\perp$ KN.

2. One of the diagonal bisects the other diagonal.
In the above kite KIND, the diagonal DI bisects the diagonal KN.
(i. e) KO = ON.

### Solved Examples

Question 1: In the given figure, ABCD is a rectangle whose diagonal AC and BD intersect at O. IF $\angle OAB$ = 32o , find
a. $\angle ACB$
b. $\angle OBC$

Solution:

$\angle ABC$ = 90o [ each angle of a rectangle = 90o ]
In $\Delta ABC$, we have
$\angle CAB$ + $\angle ABC$ + $\angle ACB$ = 180o [ sum of the angles of a triangle ]
=> 32 + 90 + $\angle ACB$ = 180o
=>                $\angle ACB$ = 180 - 90 - 32
$\angle ACB$   = 58o  - - - - - - - - - - -  - - - - ( a )

We know that the diagonals of a rectangle are congruent and bisect each other.
Therefore, OA = OB   => $\angle OBA$ = $\angle OAB$ = 32o
$\angle ABC$ = 90o  => $\angle OBA$+ $\angle OBC$ = 90o
=>              32    + $\angle OBC$ = 90
=>                         $\angle OBC$ = 90 - 32
$\angle OBC$    = 58o  - - - - - - - - - - - - - - - ( b )

Question 2: The lengths of the diagonals of a rhombus are 24 cm and 18 cm respectively. Find the length of each side of the rhombus.
Solution:

Let ABCD be the Rhombus whose diagonals AC and BD intersect at O.
Then AC = 24 cm and BD = 18 cm

Since the diagonals of a rhombus bisect each other,
we have OA = OC = 12 cm and  OB = OD = 9 cm

Since the diagonals of a rhombus bisect each other at right angles, we have $\angle AOB$ = 90o

Let the length of each side of the rhombus be x cm.
Then, in right angled $\Delta AOB$, by Pythagoras Theorem, we have,
AB2 = OA2 + OB2
=>           x2 = 122 + 92
= 144 + 81
= 225
=>             x = $\sqrt{225}$
= 15.
Hence the length of each side of the rhombus is 15 cm.