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I. Fill in the blanks
1. The sum of angles of a quadrilateral is _______ degree.
2. Every square is a ________ but not every rectangle is not a _______.
3. A _____ is a quadrilateral with four congruent sides and congruent angles.
4. A Quadrilateral has ______ sides.

Solution:

1. 360
2. rectangle, square
3. Square
4. four

Example 1.
One angle of a quadrilateral is 60°and the other three angles are equal. What is the measure of each of these equal angles?

Solution:

Given,
One angle of a quadrilateral is =60°
The other three angles are equal.
We know that in a quadrilateral, the sum of the angles of a quadrilateral is 360.
Therefore, 360-60=300
The measure of each of these equal angles=300/3=100
Hence, the measure of each of these equal angles is 100.

Example 2.
Two adjacent sides AB and BC of a parallelogram ABCD are in the ratio 3:5. If the perimeter of the parallelogram is 192 cm, What is the length of AB and BC? Solution:

Given,
Two adjacent sides AB and BC of a parallelogram ABCD are in the ratio of 3:5
Perimeter of the parallelogram=192 cm
Let the sides be 3x and 5x.
We know that opposite sides of the parallelogram are equal to each other.
5x+3x+5x+3x=192
16x=192
x=192/16=12
AB=3x=3(12)=36
BC=5x=5(12)=60
Hence, Length of AB and Bc are 36 cm, 60 cm.

Example 3.
If one angle of a parallelogram is 800. Find its opposite and adjacent angles of the parallelogram? Solution:

Given,
One angle of a parallelogram=800
We know that opposite angles of a parallelogram are equal.
So the opposite angle is also 800.
Let the other two adjacent angles be x.
We also know that the sum of all angles=3600
So 80 +80 + x+x=360
160+2x=360
2x=360-160=200
x=1000
Therefore, the opposite angle and adjacent angles are 800,1000.

Example 4.
A quadrilateral has three angles equal to 60,60, and 120. Find its fourth angle?

Solution:

Given three angles of a quadrilateral are 60,60, and 120.
Let the fourth angle be x.
We know that sum of four angles=360
60+60+120+x=360
240+x=360
x=360-240=120
Hence, the fourth angle is 120.

Example 5.
What is the area of a rhombus whose perimeter is 80 cm and one of the diagonal is 30 cm? Solution:

Given,
The perimeter of the rhombus=80 cm
one of the diagonal=30 cm
Let the sides of the rhombus AB=Bc=CD=DA=xcm
Perimeter of rhombus=x +x +x +x=4x
4x=80
x=80/4=20
Therefore, All sides of the rhombus=20 cm
BD=30 cm
Semiperimeter=20+20+30/2=70/2=35
Area=$$\sqrt{s(s-a)(s-b)(s-c) }$$
=$$\sqrt{35(35-20)(35-20)(35-30) }$$
=$$\sqrt{ 35(15)(15)(5) }$$
=75$$\sqrt{ 7}$$
Therefore area of ABCD=2 × 75$$\sqrt{ 7}$$
=150$$\sqrt{ 7}$$
Therefore, Area of the rhombus=150$$\sqrt{ 7}$$.

Example 6.
The length of the diagonals of the rhombus is 28 cm and 36 cm. Find the perimeter of the rhombus? Solution:

Given,
Length of diagonals of rhombus=28 cm, 36 cm
OC=28/2=14 cm
OD=36/2=18 cm
We will find the side of the rhombus by the Pythagoras theorem.
DOC is a right-angle triangle, we know that a2+b2=c2
142+182=c2
196+324=c2
520=c2
c=22.80
The side of the rhombus is 22.80 cm
The perimeter of rhombus=4 × side of the rhombus
=4 × 22.80
=91 cm
Hence, the perimeter of the rhombus is 91 cm.

Example 7.
In a trapezium, the measurement of one parallel side is two more than the other parallel side and the height is 2 cm. The area of the trapezium is 32 cm2. Find the lengths of the two parallel sides. Solution:

Given,
h=2 cm
Area=32 cm2
Let a and b be the two parallel sides.
One parallel side is two more than the other.
a=b+2
Area of trapezium=32 cm2
1/2(a+b)h=32
1/2(b+2+b)2=32
2b+2=32
2b=30
b=15
a=15+2=17
Hence, the lengths of the two parallel sides are 15 cm, 17cm.

Example 8.
The perimeter of a square is 36 cm. Find the area and the length of its diagonal? Solution:

Given,
The perimeter of a square=36 cm
4s=36
s=9 cm
Area=side2
=9 × 9=81 sq cm
Area of square = 1/2×(diagonal)²
81=1/2×(diagonal)²
(diagonal)²=81 × 2
(diagonal)²=162
diagonal=$$\sqrt{ 162 }$$
=12.72 cm

Example 9.
Find the number of sides of a regular polygon whose each exterior angle has a measure of 72 degrees.

Solution:

Let the number of sides of the polygon be n.
Sum of exterior angles=3600
Given that measure of each exterior angle is 72 degrees.
We know the formula, No. of sides × measure of each exterior angle=3600
n × 72=360
n=360/72=5
Therefore, no. of sides of a regular polygon is 5.

Example 10.
Five of the angles of a hexagon are each 1100. Calculate the measure of the sixth angle. Solution:

Each of the five of the angles of a hexagon is =1100
We know that sum of all angles of hexagon=720
Let the sixth angle of hexagon=x
x+110+110+110+110+110=720
x+550=720
x=720-550
x=170
Therefore, the Sixth angle is 1700

Example 11.
The interior angle of a regular polygon is five times its exterior angle. What is the number of sides?

Solution:

Let the exterior angle be x.
Then interior angle will be 5x.
The sum of interior angles+sum of exterior angles=180°
x + 5x = 180°
x = 180/6= 30°
Since the sum of exterior angles=360°. Divide that by the exterior angle and you have the number of angles, hence the number of sides.
360÷30=12
Hence, the number of sides is 12.

Example 12.
The ratio between the interior angles and exterior angles of a regular polygon is 5:1. Find the number of sides of the polygon?

Solution:

Let interior angle and exterior angle of a polygon are 5x° and x° respectively.
We know that sum of interior and exterior angles is equal to 180°.
5x+x=180°
6x=180° => x= 30° [exterior angle is 30°]
But exterior angle = 360°/n
Therefore. 360°/n=30° => n = 360°/30°. = 12 sides.

Example 13.
In the adjoining isosceles trapezium, ABCD <A is equal to 1020. Find all the remaining angles of the trapezium​? Solution:

Given,
<A=1020
Since it is an adjoining isosceles trapezium ABCD,
AB parallel to CD and BC parallel to AD