Free Exemplar Exercise 3.1 Practice Test - 8th Grade

Question 1

Question 1
If three angles of a quadrilateral are each equal to $75^{\circ}$ , the fourth angle is
a) $150^{\circ}$
b) $135^{\circ}$
c) $45^{\circ}$
d) $75^{\circ}$

SOLUTION

Solution : Given, three angles of a quadrilateral $= 75^{\circ}$
Let the fourth angle be $x^{\circ}$ .
Then, according to the property, $75^{\circ} + 75^{\circ} + 75^{\circ} + x^{\circ} = 360^{\circ}$ , since sum of the angles of a quadrilateral is $360^{\circ}$ .
So, $225^{\circ} + x^{\circ} = 360^{\circ}$ or $x^{\circ} = 360^{\circ} - 225^{\circ} = 135^{\circ}$
Hence, the fourth angle is $135^{\circ}$ .

Question 2

Question 2
For which of the following, diagonals bisect each other?
a) Square
b) Kite
c) Trapezium
d) Quadrilateral

SOLUTION

Solution : We know that the diagonals of a square bisect each other but the diagonals of a trapezium, kite and quadrilateral do not bisect each other.

Question 3

Question 3
For which of the following, all angles are equal?
a) Rectangle
b) Kite
c) Trapezium
d) Rhombus

SOLUTION

Solution : In a rectangle, all angles are equal, and each angle measures $90^{\circ}$ .

Question 4

Question 4
For which of the following, diagonals are perpendicular to each other?
a) Parallelogram
b) Kite
c) Trapezium
d) Rectangle

SOLUTION

Solution : The diagonals of a kite are perpendicular to each other.

Question 5

Question 5
For which of the following, diagonals are equal?
a) Trapezium
b) Rhombus
c) Parallelogram
d) Rectangle

SOLUTION

Solution : By the property of a rectangle, we know that its diagonals are equal.

Question 6

Question 6
Which of the following figures satisfy the following properties?
-All sides are congruent.
-All angles are right angles.
-Opposite sides are parallel.

SOLUTION

Solution : We know that all the properties mentioned above are related to a square and we can observe that figure R resembles a square.

Hence, the figure R satisfies the following properties.

Question 7

Question 7
Which of the following figures satisfy the following property?
-Has two pairs of congruent adjacent sides.

SOLUTION

Solution : We know that, a kite has two pairs of congruent adjacent sides and we can observe that figure R resembles a kite.

Question 8

Question 8
Which of the following figures satisfy the following property?
- Only one pair of sides are parallel.

SOLUTION

Solution : We know that, in a trapezium, only one pair of sides are parallel and we can observe that figure P resembles a trapezium.

Question 9

Question 9
Which of the following figures do not satisfy any of the following properties?
- All sides are equal.
-All angles are right angles.
-Opposite sides are parallel.

SOLUTION

Solution : On observing the above figures, we conclude that the figure P does not any satisfy any of the given properties.

Question 10

Question 10
Which of the following properties describe a trapezium?
a) A pair of opposite sides is parallel
b) The diagonals bisect each other
c) The diagonals are perpendicular to each other
d) The diagonals are equal

SOLUTION

Solution : We know that, in a trapezium, a pair of opposite sides are parallel.

Question 11

Question 11
Which of the following is a property of a parallelogram?
a) Opposite sides are parallel
b) The diagonals bisect each other at right angles
c) The diagonals are perpendicular to each other
d) All angles are equal

SOLUTION

Solution : We know that, in a parallelogram, opposite sides are parallel and equal.
The diagonals bisect each other but not always at right angles.
The diagonals are not perpendicular to each other in all parallelograms.
All angles are equal in all parallelograms.

Hence, option (a) is correct.

Question 12

Question 12
What is the maximum number of obtuse angles that a quadrilateral can have?
a) 1
b) 2
c) 3
d) 4

SOLUTION

Solution : We know that the sum of all the angles of a quadrilateral is $360^{\circ}$ .
Also, an obtuse angle is more than $90^{\circ}$ and less than $180^{\circ}$ .
Thus, all the angles of a quadrilateral cannot be obtuse.
Hence, at the most 3 angles can be obtuse.

Question 13

Question 13
How many non-overlapping triangles can we make in an n-gon (polygon having n sides), by joining the vertices?
a) n - 1
b) n - 2
c) n - 3
d) n - 4

SOLUTION

Solution : The number of non-overlapping triangles in an n-gon = n - 2,
i.e. 2 less than the number of sides.

Question 14

Question 14
What is the sum of all the angles of a pentagon?
a) $180^{\circ}$
b) $360^{\circ}$
c) $540^{\circ}$
d) $720^{\circ}$

SOLUTION

Solution : $We know that, the sum of angles of a polygon$
$= (n - 2) \times 180^{\circ},$
$where n is the number of sides of the polygon.$
$In pentagon, n = 5$
$∴ Sum of the angles$
$= (n - 2) \times 180^{\circ}$
$= (5 - 2) \times 180^{\circ}$
$= 3 \times 180^{\circ}$
$= 540^{\circ}$

Question 15

Question 15
What is the sum of all angles of a hexagon?
a) $180^{\circ}$
b) $360^{\circ}$
c) $540^{\circ}$
d) $720^{\circ}$

SOLUTION

Solution : $We know that, the sum of angles of a polygon$
$= (n - 2) \times 180^{\circ},$
$where n is the number of sides of the polygon.$
$In hexagon, n = 6$
$∴ Sum of the angles$
$= (n - 2) \times 180^{\circ}$
$= (6 - 2) \times 180^{\circ}$
$= 4 \times 180^{\circ}$
$= 720^{\circ}$

Question 16

Question 16
If two adjacent angles of a parallelogram are $(5 x - 5)^{\circ}$ and $(10 x + 35)^{\circ}$ , then the ratio of these angles is ___________.
a) 1 : 3
b) 2 : 3
c) 1 : 4
d) 1 : 2

SOLUTION

Solution : We know that the adjacent angles of a parallelogram are supplementary, i.e. their sum is equal to $180^{\circ}$ ,
$∴ (5 x - 5) + (10 x + 35) = 180^{\circ} \Rightarrow 15 x + 30^{\circ} = 180^{\circ} \Rightarrow 15 x = 150^{\circ} \Rightarrow x = 10^{\circ}$
Thus, the angles are $(5 \times 10 - 5) = 45^{\circ}$ and $(10 \times 10 + 35) = 135^{\circ}$ .
Hence, the required ratio is $45^{\circ} : 135^{\circ}$ , that is, 1 : 3.

Question 17

Question 17
A quadrilateral in which all sides are equal, opposite angles are equal and the diagonals bisect each other at right angles is a ___.
a) Rhombus
b) Parallelogram
c) Square
d) Rectangle

SOLUTION

Solution : We know that, in rhombus all sides are equal, opposite angles are equal and diagonals bisect each other at right angles.

Question 18

Question 18
A quadrilateral whose opposite sides and all the angles are equal is a
a) Rectangle
b) Parallelogram
c) Square
d) Rhombus

SOLUTION

Solution : We know that in a rectangle, opposite sides and all the angles are equal.

Question 19

Question 19
A quadrilateral in which all sides, diagonals and angles are equal is a
a) Square
b) Trapezium
c) Rectangle
d) Rhombus

SOLUTION

Solution : In a square, all the sides are equal in length, the diagonals are equal in length and all the angles are equal in magnitude.

Hence, (a) is the correct option.

Question 20

Question 20
How many diagonals does a hexagon have?
a) 9
b) 8
c) 2
d) 6

SOLUTION

Solution : We know that, the number of diagonals in a polygon of n sides is :
$\frac{n (n - 3)}{2}$
In hexagon, n = 6
$∴$ Number of diagonals in a hexagon $= \frac{6 (6 - 3)}{2} = \frac{6 \times 3}{2} = 3 \times 3 = 9$

Question 21

Question 21
If the adjacent sides of a parallelogram are equal, then parallelogram is a
a) Rectangle
b) Trapezium
c) Rhombus
d) Square

SOLUTION

Solution : We know that, in a parallelogram, opposite sides are equal.
But according to the question, adjacent sides are also equal.
Thus, the parallelogram in which all the sides are equal is known as the rhombus.

Question 22

Question 22
If the diagonals of a quadrilateral are equal and bisect each other, then the quadrilateral is a
a) Rhombus
b) Rectangle
c) Trapezium
d) Parallelogram

SOLUTION

Solution : In a rectangle, the diagonals are equal and bisect each other.

Hence, (b) is the correct option.

Question 23

Question 23
The sum of all exterior angles of a triangle is
a) $180^{\circ}$
b) $360^{\circ}$
c) $540^{\circ}$
d) $720^{\circ}$

SOLUTION

Solution : We know that the sum of all exterior angles of any polygon is $360^{\circ}$ .
Triangle is also a polygon. Hence, the sum of all exterior angles of a triangle is $360^{\circ}$ .

Question 24

Question 24
Which of the following is an equiangular and equilateral polygon?
a) Square
b) Rectangle
c) Rhombus
d) Right angled Triangle

SOLUTION

Solution : In a square, all the sides and all the angles are equal.
Hence, square is an equiangular and equilateral polygon.

Question 25

Question 25
Which one has all the properties of a kite and a parallelogram?
a) Trapezium
b) Rhombus
c) Rectangle
d) Parallelogram

SOLUTION

Solution : Properties of a kite:
Two pairs of equal sides.
Diagonals bisect at $90^{\circ}$ .
One pair of opposite angles are equal.

Properties of a parallelogram:
Opposite sides are equal.
Opposite angles are equal.
Diagonals bisect each other.

So, from the given options, all these properties are satisfied by the rhombus.

Question 26

Question 26
The angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4. The smallest angle is
a) $72^{\circ}$
b) $144^{\circ}$
c) $36^{\circ}$
d) $18^{\circ}$

SOLUTION

Solution : Given, the angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4.
Let the angles of the given quadrilateral be $x^{\circ}, 2 x^{\circ}, 3 x^{\circ}$ and $4 x^{\circ}$ .
$∴ x^{\circ} + 2 x^{\circ} + 3 x^{\circ} + 4 x^{\circ} = 360^{\circ}$ [ sum of the angles of a quadrilateral is $360^{\circ}$ ]
$\Rightarrow 10 x = 360^{\circ} \Rightarrow x = \frac{360^{\circ}}{10} = 36^{\circ}$
Hence, the smallest angle is $36^{\circ}$ .

Question 27

Question 27
In the trapezium ABCD, the measure of $∠ D$ is
a) $55^{\circ}$
b) $115^{\circ}$
c) $135^{\circ}$
d) $125^{\circ}$

SOLUTION

Solution : We know that, in a trapezium, the angles on either side of the base are supplementary.
In trapezium ABCD,
$∴ ∠ A + ∠ D = 180^{\circ} \Rightarrow 55^{\circ} + ∠ D = 180^{\circ} \Rightarrow ∠ D = 180^{\circ} - 55^{\circ} \Rightarrow ∠ D = 125^{\circ}$

Question 28

Question 28
A quadrilateral has three acute angles. If each measures $80^{\circ}$ , then the measure of the fourth angle is :
a) $150^{\circ}$
b) $120^{\circ}$
c) $105^{\circ}$
d) $140^{\circ}$

SOLUTION

Solution : Let the fourth angle be x, then $80^{\circ} + 80^{\circ} + 80^{\circ} + x = 360^{\circ}$
[ $∵$ sum of all the angles of quadrilaterals is $360^{\circ}$ ]
$\Rightarrow 240^{\circ} + x = 360^{\circ} \Rightarrow x = 360^{\circ} - 240^{\circ} \Rightarrow x = 120^{\circ}$

Question 29

Question 29
The number of sides of a regular polygon in which each exterior angle has a measure of $45^{\circ}$ is
a) 8
b) 10
c) 4
d) 6

SOLUTION

Solution : We know that the sum of all exterior angles of a polygon is $360^{\circ}$ .
Since each exterior angle measures $45^{\circ}$ , the number of sides
$= \frac{S u m o f a l l e x t e r i o r a n g l e s}{M e a s u r e o f a n e x t e r i o r a n g l e} = \frac{360^{\circ}}{45^{\circ}} = 8$

Question 30

Question 30
In a parallellogram PQRS, if $∠ P = 60^{\circ}$ , then other three angles are
a) $45^{\circ}, 135^{\circ}, 120^{\circ}$
b) $60^{\circ}, 120^{\circ}, 120^{\circ}$
c) $60^{\circ}, 135^{\circ}, 135^{\circ}$
d) $45^{\circ}, 135^{\circ}, 135^{\circ}$

SOLUTION

Solution : Given, $∠ P = 60^{\circ}$

Since, in a parallelogram, adjacent angles are supplementary,
$∠ P + ∠ Q = 180^{\circ} \Rightarrow 60^{\circ} + ∠ Q = 180^{\circ} \Rightarrow ∠ Q = 120^{\circ}$
Also, opposite angles are equal in a parallelogram.
Therefore, $∠ R = ∠ P = 60^{\circ}, ∠ S = ∠ Q = 120^{\circ}$
Hence, other three angles are $60^{\circ}, 120^{\circ}, 120^{\circ}$ .

Question 31

Question 31
If two adjacent angles of a parallelogram are in the ratio 2 : 3, then the measures of the angles are
a) $72^{\circ}, 108^{\circ}$
b) $36^{\circ}, 54^{\circ}$
c) $80^{\circ}, 120^{\circ}$
d) $96^{\circ}, 144^{\circ}$

SOLUTION

Solution : Given, two adjacent angles of a parallelogram are in the ratio 2 : 3
Let the angles be 2x and 3x.
Then,
$2 x + 3 x = 180^{\circ}$ [ $∵$ adjacent angles of a parallelogram are supplementary]
$\Rightarrow 5 x = 180^{\circ} \Rightarrow x = 36^{\circ}$
Hence, the measures of the angles are $2 x = 2 \times 36^{\circ} = 72^{\circ}$ and $3 x = 3 \times 36^{\circ} = 108^{\circ}$ .

Question 32

Question 32
If PQRS is a parallelogram, then $∠ P - ∠ R$ is equal to
a) $60^{\circ}$
b) $90^{\circ}$
c) $80^{\circ}$
d) $0^{\circ}$

SOLUTION

Solution : In a parallelogram, opposite angles are equal. Therefore, $∠ P - ∠ R = 0$ , as $∠ P$ and $∠ R$ are opposite angles.

Question 33

Question 33
The sum of adjacent angles of a parallelogram is
a) $180^{\circ}$
b) $120^{\circ}$
c) $360^{\circ}$
d) $90^{\circ}$

SOLUTION

Solution : By property of the parallelogram, we know that, the sum of adjacent angles of a parallelogram is $180^{\circ}$ .

Question 34

Question 34
The angle between the two altitudes of a parallelogram through the same vertex of an obtuse angle of the parallelogram is $30^{\circ}$ . The measure of the obtuse angle is
a) $100^{\circ}$
b) $150^{\circ}$
c) $105^{\circ}$
d) $120^{\circ}$

SOLUTION

Solution : Let EC and FC be altitudes and $∠ E C F = 30^{\circ}$ .

Let $∠ E D C = x = ∠ F B C$
So, $∠ E C D = 90 - x^{\circ} a n d ∠ B C F = 90^{\circ} - x$
So, by property of the parallelogram,
$∠ A D C + ∠ D C B = 180^{\circ} ∠ A D C + (∠ E C D + ∠ E C F + ∠ B C F) = 180^{\circ} \Rightarrow x + 90^{\circ} - x + 30^{\circ} + 90^{\circ} - x = 180^{\circ} \Rightarrow - x = 180^{\circ} - 210^{\circ} = - 30^{\circ} \Rightarrow x = 30^{\circ}$
Hence, $∠ D C B = 30^{\circ} + 60^{\circ} + 60^{\circ} = 150^{\circ}$ .

Question 35

Question 35
In the given figure, ABCD and BDCE are parallelograms with common base DC. If $B C ⊥ B D$ , then $∠ B E C$ is equal to
a) $60^{\circ}$
b) $30^{\circ}$
c) $150^{\circ}$
d) $120^{\circ}$

SOLUTION

Solution : $∠ B A D = 30^{\circ}$ [given]
$∴$ $∠ B C D = 30^{\circ}$ [ $∵$ opposite angles of a parallelogram are equal]
In $Δ C B D$ , by angle sum property of a traingle, we have,
$∠ D B C + ∠ B C D + ∠ C D B = 180^{\circ} \Rightarrow 90^{\circ} + 30^{\circ} + ∠ C D B = 180^{\circ} \Rightarrow ∠ C D B = 180^{\circ} - 120^{\circ} = 60^{\circ}$
$∴ ∠ B E C = ∠ C D B = 60^{\circ}$ [ $∵$ opposite angles of a parallelogram are equal]

Question 36

Question 36
Length of one of the diagonals of a rectangle whose sides are 10 cm and 24 cm is
a) 25 cm
b) 20 cm
c) 26 cm
d) 3.5 cm

SOLUTION

Solution :
In $Δ B C D$ ,
$∠ B D C = 90^{\circ}$
$∴$ Using Pythagoras theorem,
$B C^{2} = B D^{2} + C D^{2}$
$\Rightarrow B C^{2} = 10^{2} + 24^{2} = 100 + 576 \Rightarrow B C^{2} = 676 \Rightarrow B C = \sqrt{676} \Rightarrow B C = 26 c m$

Question 37

Question 37
If the adjacent angles of a parallelogram are equal, then the parallelogram is a
a) Rectangle
b) Trapezium
c) Rhombus
d) None of these

SOLUTION

Solution : We know that the adjacent angles of a parallelogram are supplementary, i.e. their sum equals $180^{\circ}$ . If they are equal, then each angle will be of measure :
$\frac{180^{\circ}}{2} = 90^{\circ}$ .
Hence, such a parallelogram is a rectangle.

Question 38

Question 38
Which of the following can be four interior angles of a quadrilateral?
a) $140^{\circ}, 40^{\circ}, 20^{\circ}, 160^{\circ}$
b) $270^{\circ}, 150^{\circ}, 30^{\circ}, 20^{\circ}$
c) $40^{\circ}, 70^{\circ}, 90^{\circ}, 60^{\circ}$
d) $110^{\circ}, 40^{\circ}, 30^{\circ}, 180^{\circ}$

SOLUTION

Solution : We know that the sum of interior angles of a quadrilateral is $360^{\circ}$
Thus, the angles in option (a) can be four interior angles of a quadrilateral as their sum is $360^{\circ}$ .

Question 39

Question 39
The sum of angles of a concave quadrilateral is
a) More than $360^{\circ}$
b) Less than $360^{\circ}$
c) Equal to $360^{\circ}$
d) Twice of $360^{\circ}$

SOLUTION

Solution : We know that the sum of interior angles of any polygon (convex or concave) having n sides is $(n - 2) \times 180^{\circ}$ .
$∴$ The sum of angles of a concave quadrilateral is $(4 - 2) \times 180^{\circ} = 360^{\circ}$ .

Question 40

Question 40
Which of the following can never be the measure of the exterior angle of a regular polygon?
a) $22^{\circ}$
b) $36^{\circ}$
c) $45^{\circ}$
d) $30^{\circ}$

SOLUTION

Solution : We know that the sum of all exterior angles of a polygon is $360^{\circ}$ .
The measure of each exterior angle is $\frac{360^{\circ}}{n}$ , where n is the number of sides/angles.
Thus, the measure of each exterior angle will always be a factor of $360^{\circ}$ .
Hence, $22^{\circ}$ can never be the measure of the exterior angle of a regular polygon.

Question 41

Question 41
In the figure, BEST is a rhombus, then the value of y - x is
a) $40^{\circ}$
b) $50^{\circ}$
c) $20^{\circ}$
d) $10^{\circ}$

SOLUTION

Solution : Given: BEST is a rhombus, so,
$T S ∥ B E$
and BS is a transversal.
$∴ ∠ S B E = ∠ T S B = 40^{\circ}$ [alternate interior anges]
$A l s o, ∠ y = 90^{\circ}$ [diagonals of a rhombus bisect at $90^{\circ}$ ]

In $Δ T S O$ ,
$∠ S T O + ∠ T S O + ∠ S O T = 180^{\circ}$ [Angle sum property of a triangle]
$x + 40^{\circ} + 90^{\circ} = 180^{\circ} \Rightarrow x = 180^{\circ} - 90^{\circ} - 40^{\circ} = 50^{\circ}$
$∴ y - x = 90^{\circ} - 50^{\circ} = 40^{\circ}$

Question 42

Question 42
The closed curve which is also a non-intersecting polygon, is:
a)

b)

c)

d)

SOLUTION

Solution : Figure (a) is a non-intersecting polygon as no two line segments intersect each other.

Question 43

Question 43
Which of the following is not true for an exterior angle of a regular polygon with n sides?
a) Each exterior angle $= \frac{360^{\circ}}{n}$
b) Exterior angle $= 180^{\circ}$ - Interior angle
c) $n = \frac{360^{\circ}}{E x t e r i o r a n g l e}$
d) Each exterior angle $= \frac{(n - 2) \times 180^{\circ}}{n}$

SOLUTION

Solution : We know that, (a) and (b) are the formulae to find the measure of each exterior angle, when the number of sides and measure of an interior angle respectively are given and (c) is the formula to find the number of sides of the polygon when an exterior angle is given.
Hence, the formula given in option (d) is not true for an exterior angle of a regular polygon with n sides.

Question 44

Question 44
PQRS is a square. PR and SQ intersect at O. Then, $∠ P O Q$ is a
a) Right angle
b) Straight angle
c) Reflex angle
d) Complete angle

SOLUTION

Solution :
We know that the diagonals of a square intersect each other at right angles.
Hence,
$∠ P O Q = 90^{\circ}$ .

Question 45

Question 45
Two adjacent angles of a parallelogram are in the ratio 1 : 5. Then, all the angles of the parallelogram are
a) $30^{\circ}, 150^{\circ}, 30^{\circ}, 150^{\circ}$
b) $85^{\circ}, 95^{\circ}, 85^{\circ}, 95^{\circ}$
c) $45^{\circ}, 135^{\circ}, 45^{\circ}, 135^{\circ}$
d) $30^{\circ}, 180^{\circ}, 30^{\circ}, 180^{\circ}$

SOLUTION

Solution : Let the adjacent angles of a parallelogram be x and 5x, respectively.
Then, $x + 5 x = 180^{\circ}$ [ $∵$ adjacent angles of a parallelogram are supplementary]
$\Rightarrow 6 x = 180^{\circ}$
$\Rightarrow x = 30^{\circ}$
$∴$ The adjacent anlges are $30^{\circ}$ and $150^{\circ}$ .
Hence, the angles are $30^{\circ}, 150^{\circ}, 30^{\circ}, 150^{\circ}$ . [ $∵$ opposite angles are equal]

Question 46

Question 46
A parallelogram PQRS is constructed with sides QR = 6 cm, PQ = 4 cm and $∠ P Q R = 90^{\circ}$ . Then, PQRS is a
a) Square
b) Rectangle
c) Rhombus
d) Trapezium

SOLUTION

Solution : We know that, if in a parallelogram one angle is of $90^{\circ}$ , then all angles will be of $90^{\circ}$ and a parallelogram with all angles equal to $90^{\circ}$ is called a rectangle.

Question 47

Question 47
The angles P, Q, R and S of a quadrilateral are in the ratio 1 : 3 : 7 : 9. Then, PQRS is a
a) Parallelogram
b) Trapezium with $P Q ∥ R S$
c) Trapezium with $Q R ∥ P S$
d) Kite

SOLUTION

Solution :

Let the angles be x, 3x, 7x and 9x, then,
$x + 3 x + 7 x + 9 x = 360^{\circ}$ [ $∵$ sum of angles in any quadrilateral is $360^{\circ}$ ]
$\Rightarrow 20 x = 360^{\circ} \Rightarrow x = \frac{360^{\circ}}{20} \Rightarrow x = 18^{\circ}$
Then, the angles P, Q, R and S are $18^{\circ}, 54^{\circ}, 126^{\circ}$ and $162^{\circ}$ , respectively.
Since, $∠ P + ∠ S = 18^{\circ} + 162^{\circ} = 180^{\circ}$ and $∠ Q + ∠ R = 54^{\circ} + 126^{\circ} = 180^{\circ}$ ,
$S o, P Q ∥ S R$ .
$∴$ The quadrilateral PQRS is a trapezium with $P Q ∥ S R$ .

Question 48

Question 48
PQRS is a trapezium in which $P Q ∥ S R$ and $∠ P = 130^{\circ}, ∠ Q = 110^{\circ}$ . Then, $∠ R$ is equal to
a) $70^{\circ}$
b) $50^{\circ}$
c) $65^{\circ}$
d) $55^{\circ}$

SOLUTION

Solution : Since, PQRS is a trapezium and $P Q ∥ S R$ .
$∴ ∠ Q + ∠ R = 180^{\circ}$ [ $∵$ angles between the pair of parallel sides are supplementary]
$\Rightarrow ∠ R = 180^{\circ} - 110^{\circ} = 70^{\circ}$

Question 49

Question 49
The number of sides of a regular polygon in which each interior angle measures $135^{\circ}$ is
a) 6
b) 7
c) 8
d) 9

SOLUTION

Solution : We know that, the measure of each exterior angle of a polygon having n sides is given by $\frac{360^{\circ}}{n}$ .

$∴$ The number of sides,
$n = \frac{360^{\circ}}{E x t e r i o r a n g l e} = \frac{360^{\circ}}{180^{\circ} - 135^{\circ}}$

[ $∵$ exterior angle + interior angle $= 180^{\circ}$ ]

$= \frac{360^{\circ}}{45^{\circ}} = 8$

Question 50

Question 50
If a diagonal of a quadrilateral bisects both the angles, then it is a
a) Kite
b) Parallelogram
c) Rhombus
d) Rectangle

SOLUTION

Solution : If a diagonal of a quadrilateral bisects both the angles, then it is a rhombus.

Question 51

Question 51
To construct a unique parallelogram, the minimum number of measurements required is
a) 2
b) 3
c) 4
d) 5

SOLUTION

Solution : We know that, in a parallelogram, opposite sides are equal and parallel. Also, opposite angles are equal.
So, to construct a parallelogram uniquely, we require the measure of any two non-parallel sides and the measure of an angle.
Hence, the minimum number of measurements required to construct a unique parallelogram is 3.

Question 52

Question 52
To construct a unique rectangle, the minimum number of measurements required is
(a) 4
(b) 3
(c) 2
(d) 1

SOLUTION

Solution : Since, in a rectangle, opposite sides are equal and parallel, so we need the measurement of only two adjacent sides, i.e. length and breadth. Also, each angle measures $90^{\circ}$ .
Hence, we require only two measurements to construct a unique rectangle.

Question 53

Question 53
Fill in the blanks to make the statements true.
In a quadrilateral HOPE, the pairs of opposite sides are ___.

SOLUTION

Solution : EH, PO and HO, EP are pairs of opposite sides.

Question 54

Question 54
Fill in the blanks to make the statements true.
In quadrilateral ROPE, the pairs of adjacent angles are ___.

SOLUTION

Solution :
The pairs of adjacent angles are $∠ R, ∠ O; ∠ O, ∠ P; ∠ P, ∠ E; ∠ E, ∠ R$ .

Question 55

Question 55
Fill in the blanks to make the statements true.
In quadrilateral WXYZ, the pairs of opposite angles are ___.

SOLUTION

Solution :
The pairs of opposite angles are $∠ W, ∠ Y$ and $∠ X, ∠ Z$ .

Question 56

Question 56
Fill in the blanks to make the statements true.
The diagonals of the quadrilateral DEFG are ___ and ___.

SOLUTION

Solution :
The diagonals are GE and FD.

Question 57

Question 57
Fill in the blanks to make the statements true.
The sum of all ___ of a quadrilateral is $360^{\circ}$ .

SOLUTION

Solution : Angles
We know that the sum of all angles of a quadrilateral is $360^{\circ}$ .

Question 58

Question 58
Fill in the blanks to make the statements true.
The measure of each exterior angle of a regular pentagon is ___.

SOLUTION

Solution : $72^{\circ}$
Measure of exterior angle $= \frac{360^{\circ}}{N u m b e r o f s i d e s} = \frac{360^{\circ}}{5} = 72^{\circ}$ . [ $∵$ in pentagon, number of sides, n = 5]

Question 59

Question 59
Fill in the blanks to make the statements true.
Sum of the angles of a hexagon is ___.

SOLUTION

Solution : $720^{\circ}$

The sum of angles of an n-gon (a polygon with n sides)
$= (n - 2) \times 180^{\circ}$ .
$∴$ Sum of the angles of a hexagon $= (6 - 2) \times 180^{\circ} = 4 \times 180^{\circ} = 720^{\circ}$ . [ $∵$ In a hexagon, number of sides, n = 6]

Question 60

Question 60
Fill in the blanks to make the statements true.
The measure of each exterior angle of a regular polygon of 18 sides is ___.

SOLUTION

Solution : $20^{\circ}$
We know that, measure of each exterior angle $= \frac{360^{\circ}}{N u m b e r o f s i d e s} = \frac{360^{\circ}}{18} = 20^{\circ}$ .

Question 61

Question 61
Fill in the blanks to make the statements true.
The number of sides of a regular polygon, where each exterior angle has a measure of $36^{\circ}$ , is ___.

SOLUTION

Solution : We know that the sum of exterior angles of a regular polygon is $360^{\circ}$ .
Given, the measure of each exterior angle = $36^{\circ}$

Number of sides
$= \frac{360^{\circ}}{E x t e r i o r a n g l e}$

$= \frac{360^{\circ}}{36^{\circ}} = 10$ .

Question 62

Question 62
Fill in the blanks to make the statements true.
is a closed curve entirely made up of line segments. The another name for this shape is ___.

SOLUTION

Solution : Concave polygon
As one interior angle is of greater than $180^{\circ}$ .

Question 63

Question 63
Fill in the blanks to make the statements true.
A quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure is ___.

SOLUTION

Solution : Kite
By the property of a kite, we know that it has two opposite angles of equal measure.

Question 64

Question 64
Fill in the blanks to make the statements true.

The measure of each angle of a regular pentagon is .

SOLUTION

Solution : A regular pentagon has 5 sides.

We know that, the sum of interior angles of a polygon
$= (n - 2) \times 180^{\circ}$

$∴$ Measure of each interior angle $= \frac{S u m o f i n t e r i o r a n g l e s}{N u m b e r o f s i d e s} = \frac{(5 - 2) \times 180^{\circ}}{5} = 3 \times 36^{\circ} = 108^{\circ}$

Question 65

Question 65
Fill in the blanks to make the statements true.
The name of a three-sided regular polygon is ___.

SOLUTION

Solution : Equilateral triangle.

Question 66

Question 66
Fill in the blanks to make the statements true.
The number of diagonals in a hexagon is ___.

SOLUTION

Solution : We know that, number of diagonals of a n-gon $= \frac{n (n - 3)}{2}$ .
Here, $n = 6$ , therefore the number of diagonals $= \frac{6 (6 - 3)}{2} = \frac{6 \times 3}{2} = 9$ .

Question 67

Question 67
Fill in the blanks to make the statements true.
A polygon is a simple closed curve made up of only ___.

SOLUTION

Solution : Line segments.

A closed curve made up of only line segments is called a polygon.

Question 68

Question 68
Fill in the blanks to make the statements true.
A regular polygon is a polygon whose all sides are equal and all ___ are equal.

SOLUTION

Solution : Angles
In a regular polygon, all sides are equal and all angles are equal.

Question 69

Question 69
Fill in the blanks to make the statements true.
The sum of interior angles of a polygon of n sides is ___ right angles.

SOLUTION

Solution : (2n - 4)
By the formula, sum of interior angles of a polygon of n sides $= (n - 2) \times 180^{\circ}$
$= (2 n - 4) \times 90^{\circ}$ .

Question 70

Question 70
Fill in the blanks to make the statements true.

The sum of all exterior angles of a polygon is .

SOLUTION

Solution : $360^{\circ}$

The sum of all exterior angles of a polygon is always $360^{\circ}$ .

Question 71

Question 71
___ is a regular quadrilateral.

SOLUTION

Solution : Square
Since in square, all the sides are of equal length and all angles are equal.

Question 72

Question 72
A quadrilateral in which a pair of opposite sides is parallel is ___.

SOLUTION

Solution : Trapezium
We know, that, in a trapezium, one pair of sides is parallel.

Question 73

Question 73
Fill in the blanks to make the statements true.

If all sides of a quadrilateral are equal, it is a .

SOLUTION

Solution : Rhombus or a Square

The length of all the sides are equal in a Rhombus or a Square

Question 74

Question 74
Fill in the blanks to make the statements true.
In a rhombus, diagonals intersect at ___ angles.

SOLUTION

Solution : Right
The diagonals of a rhombus intersect at right angles.

Question 75

Question 75

___ measurements can determine a quadrilateral uniquely.

SOLUTION

Solution : 5

To construct a unique quadrilateral, we require 5 measurements, i.e. four sides and one angle or three sides and two included angles or two adjacent sides and three angles should be given.

Question 76

Question 76
Fill in the blanks to make the statements true.
A quadrilateral can be constructed uniquely if its three sides and ___ angles are given.

SOLUTION

Solution : Two included
We can determine a quadrilateral uniquely if three sides and two included angles are given.

Question 77

Question 77
Fill in the blanks to make the statements true.

A rhombus is a parallelogram in which ___ sides are equal.

SOLUTION

Solution : all

A rhombus is a parallelogram in which all sides are equal.

Question 78

Question 78
Fill in the blanks to make the statements true.
The measure of ___ angle of a concave quadrilateral is more than $180^{\circ}$ .

SOLUTION

Solution : One
A concave polygon is a polygon in which at least one interior angle is more than $180^{\circ}$ .

Question 79

Question 79
Fill in the blanks to make the statements true.

A diagonal of a quadrilateral is a line segment that joins two ___ vertices of the quadrilateral.

SOLUTION

Solution : non-adjacent / opposite

The line segment connecting two non-adjacent / opposite vertices is called diagonal.

Question 80

Question 80
Fill in the blanks to make the statements true.
The number of sides in a regular polygon having the measure of an exterior angle as $72^{\circ}$ is ___.

SOLUTION

Solution : 5
We know that,
the sum of exterior angles of any polygon is $360^{\circ}$ .
The number of sides in a regular polygon $= \frac{360^{\circ}}{E x t e r i o r a n g l e}$ .
$∴$ The number of sides in given polygon $= \frac{360^{\circ}}{72^{\circ}} = 5$ .

Question 81

Question 81
Fill in the blanks to make the statements true.
If the diagonals of a quadrilateral bisect each other, it is a ___.

SOLUTION

Solution : Parallelogram
In a parallelogram, the diagonals bisect each other.

Question 82

Question 82
Fill in the blanks to make the statements true.
The adjacent sides of a parallelogram are 5 cm and 9 cm. Its perimeter is ___.

SOLUTION

Solution : 28 cm
Perimeter of a parallelogram = 2(Sum of lengths of adjacent sides)
$= 2 (5 + 9) = 2 \times 14 = 28 c m$ .

Question 83

Question 83
Fill in the blanks to make the statements true.
A nonagon has ___ sides.

SOLUTION

Solution : 9
Nonagon is a polygon which has 9 sides.

Question 84

Question 84
Fill in the blanks to make the statements true.
Diagonals of a rectangle are ___.

SOLUTION

Solution : Equal
We know that, in a rectangle, both the diagonals are of equal length.

Question 85

Question 85
Fill in the blanks to make the statements true.
A polygon having 10 sides is known as ___.

SOLUTION

Solution : Decagon
A polygon with 10 sides is called decagon.

Question 86

Question 86
Fill in the blanks to make the statements true.
A rectangle whose adjacent sides are equal becomes a ___.

SOLUTION

Solution : Square
If in a rectangle, adjacent sides are equal, then it is called a square.

Question 87

Question 87
Fill in the blanks to make the statements true.
If one diagonal of a rectangle is 6 cm long, length of the other diagonal is ___.

SOLUTION

Solution : 6 cm
Since both the diagonals of a rectangle are equal. Therefore, length of other diagonal is also 6 cm.

Question 88

Question 88
Fill in the blanks to make the statements true.
Adjacent angles of a parallelogram are ___

SOLUTION

Solution : Supplementary
By property of a parallelogram, we know that the adjacent angles of a parallelogram are supplementary.

Question 89

Question 89
Fill in the blanks to make the statements true.

If only one diagonal of a quadrilateral bisects the other, then the quadrilateral is known as .

SOLUTION

Solution : Kite

In a kite, one diagonal of a the quadrilateral bisects the other.

Question 90

Question 90
Fill in the blanks to make the statements true.
In trapezium ABCD with $A B ∥ C D$ , if $∠ A = 100^{\circ}$ , then $∠ D =$ ___.

SOLUTION

Solution : $80^{\circ}$

In a trapezium, we know that the angles on either side of the base are supplementary.
So, in trapezium ABCD, given $A B ∥ C D$
$∴ ∠ A + ∠ D = 180^{\circ} \Rightarrow 100^{\circ} + ∠ D = 180^{\circ} \Rightarrow ∠ D = 180^{\circ} - 100^{\circ} ∴ ∠ D = 80^{\circ}$ .

Question 91

Question 91
Fill in the blanks to make the statements true.
The polygon in which sum of all exterior angles is equal to the sum of interior angles is called ___.

SOLUTION

Solution : Quadrilateral
We know that the sum of exterior angles of a polygon is $360^{\circ}$ and in a quadrilateral, the sum of interior angles is also $360^{\circ}$ . Therefore, a quadrilateral is a polygon in which the sum of both interior and exterior angles are equal.

Question 92

Question 92
State whether the statements are True or False.
All angles of a trapezium are equal.

SOLUTION

Solution : False
All angles of a trapezium are not equal.

Question 93

Question 93
State whether the statements are True or False.
All squares are rectangles.

SOLUTION

Solution : True.
A square a quadrilateral with all four angles right angles. Therefore, we can say that all squares are rectangles but vice-versa is not true.

Question 94

Question 94
State whether the statements are True or False.
All kites are squares.

SOLUTION

Solution : False
Kites do not satisfy all the properties of a square.
e.g. In a square, all the angles measure $90^{\circ}$ but in a kite, the measures of angles can vary.

Question 95

Question 95
State whether the statements are True or False.
All rectangles are parallelograms.

SOLUTION

Solution : True
Rectangles are special parallelograms and satisfy all properties of parallelograms. Therefore, we can say that all rectangles are parallelograms but vice-versa is not true.

Question 96

Question 96
State whether the statements are True or False.
All rhombuses are squares.

SOLUTION

Solution : False
In a rhombus, each angle is not a right angle, so rhombuses are not squares.

Question 97

Question 97
State whether the statements are True or False.
Sum of all the angles of a quadrilateral is $180^{\circ}$ .

SOLUTION

Solution : False
The sum of all the angles of a quadrilateral is $360^{\circ}$ .

Question 98

Question 98
State whether the statements are True or False.
A quadrilateral has two diagonals.

SOLUTION

Solution : True

A quadrilateral has two diagonals.

Question 99

Question 99
State whether the statements are True or False.
Triangle is a polygon whose sum of exterior angles is double the sum of interior angles.

SOLUTION

Solution : True
The sum of interior angles of a triangle is $180^{\circ}$ and the sum of exterior angles is $360^{\circ}$ , which is twice the sum of interior angles.

Question 100

Question 100
State whether the statements are True or False.

is a polygon.

SOLUTION

Solution : True
Although the figure is not a simple closed curve, it is made up of line segments and is a self-intersecting polygon.

Question 101

Question 101
State whether the statements are True or False.
A kite is not a convex quadrilateral.

SOLUTION

Solution : False
A kite is a convex quadrilateral since it has no external diagonals.

Question 102

Question 102
State whether the statements are True or False.
The sum of interior angles and the sum of exterior angles taken in an order are equal in case of quadrilaterals only.

SOLUTION

Solution : True
The sum of interior angles as well as the sum of exterior angles of a quadrilateral is equal to $360^{\circ}$ .

Question 103

Question 103
State whether the statements are True or False.
If the sum of interior angles is double the sum of exterior angles taken in an order of a polygon, then it is a hexagon.

SOLUTION

Solution : True
The sum of exterior angles of a hexagon is $360^{\circ}$ . The sum of interior angles of a hexagon is $720^{\circ}$ , which is double the sum of exterior angles.

Question 104

Question 104
State whether the statements are True or False.
A polygon is regular if all of its sides are equal.

SOLUTION

Solution : False
By definition of a regular polygon, we know that a polygon is regular if all sides and all angles are equal.

Question 105

Question 105
State whether the statements are True or False.
The rectangle is a regular quadrilateral.

SOLUTION

Solution : False
In a rectangle, all sides are not equal, hence it is not a regular polygon.

Question 106

Question 106
State whether the statements are True or False.
If diagonals of a quadrilateral are equal, it must be a rectangle.

SOLUTION

Solution : False
In a rectangle, diagonals are equal, but every quadrilateral having equal diagonals need not be a rectangle. For example, a trapezium can also have equal diagonals.

Question 107

Question 107
State whether the statements are True or False.
If opposite angles of a quadrilateral are equal, it must be a parallelogram.

SOLUTION

Solution : True
If opposite angles are equal, it has to be a parallelogram.

Question 108

Question 108
State whether the statements are True or False.
The interior angles of a triangle are in the ratio 1 : 2 : 3, then the ratio of its exterior angles is 3 : 2 : 1.

SOLUTION

Solution : False
Given, the ratio of interior angles = 1 : 2 : 3.
Let the interior angles be x, 2x and 3x.
So, $x + 2 x + 3 x = 180^{\circ}$ [angle sum property of triangle]
$\Rightarrow 6 x = 180^{\circ} \Rightarrow x = \frac{180^{\circ}}{6} = 30^{\circ}$
$∴$ The interior angles are $30^{\circ}, 60^{\circ}$ and $90^{\circ}$ .
Now, the exterior angles will be $(180^{\circ} - 30^{\circ}), (180^{\circ} - 60^{\circ})$ and $(180^{\circ} - 90^{\circ})$
i.e. $150^{\circ}, 120^{\circ}$ and $90^{\circ}$ .
The ratio of exterior angles $= 150^{\circ} : 120^{\circ} : 90^{\circ} = 15 : 12 : 9 = 5 : 4 : 3$ .

Question 109

Question 109
State whether the statements are True or False.
is a concave pentagon.

SOLUTION

Solution : False
It has 6 sides, therefore it is a concave hexagon.

Question 110

Question 110
State whether the statements are True or False.
Diagonals of a rhombus are equal and perpendicular to each other.

SOLUTION

Solution : False
Diagonals of a rhombus are perpendicular to each other but not equal.

Question 111

Question 111
State whether the statements are True or False.
Diagonals of a rectangle are equal.

SOLUTION

Solution : True
The diagonals of a rectangle are equal.

Question 112

Question 112
State whether the statements are True or False.
Diagonals of a rectangle bisect each other at right angles.

SOLUTION

Solution : False
Diagonals of a rectangle do not bisect each other at right angles.

Question 113

Question 113
State whether the statements are True or False.
Every kite is a parallelogram.

SOLUTION

Solution : False
Kite is not a parallelogram as its opposite sides are not equal and parallel.

Question 114

Question 114
State whether the statements are True or False.
Every trapezium is a parallelogram.

SOLUTION

Solution : False
In a trapezium, only one pair of sides is parallel.

Question 115

Question 115
State whether the statements are True or False.
Every parallelogram is a rectangle.

SOLUTION

Solution : False
In a parallelogram, all angles are not right angles, while in a rectangle, all angles are equal and are right angles.

Question 116

Question 116
State whether the statements are True or False.
Every trapezium is a rectangle.

SOLUTION

Solution : False
Since in a rectangle, all angles measure $90^{\circ}$ , but in a trapezium, the measure of the angles can vary.

Question 117

Question 117
State whether the statements are True or False.
Every rectangle is a trapezium.

SOLUTION

Solution : False
A trapezium is a quadrilateral that has exactly one pair of parallel sides.
A rectangle is a parallelogram with one angle equal to 90 $^{\circ}$ .

Question 118

Question 118
State whether the statements are True or False.
Every square is a rhombus.

SOLUTION

Solution : True
A square possesses all the properties of a rhombus. So, we can say that every square is a rhombus but vice-versa is not true.

Question 119

Question 119
State whether the statements are True or False.
Every square is a parallelogram.

SOLUTION

Solution : True
Every square is also a parallelogram as it has all the properties of a parallelogram but vice-versa is not true.

Question 120

Question 120
State whether the statements are True or False.
Every square is a trapezium.

SOLUTION

Solution : False.
A trapezium is a quadrilateral with exactly one pair of parallel sides.
A square is a parallelogram with all sides equal and one angle equal to $90^{\circ}$ .

Question 121

Question 121
State whether the statements are True or False.
Every rhombus is a trapezium.

SOLUTION

Solution : False
A trapezium is a quadrilateral with exactly one pair of parallel sides.
A rhombus is a parallelogram having all sides equal.

Question 122

Question 122
State whether the statements are True or False.
A quadrilateral can be drawn if only measures of four sides are given.

SOLUTION

Solution : False
We require at least five measurements to construct a quadrilateral.

Question 123

Question 123
State whether the statements are True or False.
A quadrilateral can have all four angles as obtuse.

SOLUTION

Solution : False
If all angles are obtuse, then their sum will exceed $360^{\circ}$ . This is not possible in case of a quadrilateral.

Question 124

Question 124
State whether the statements are True or False.
A quadrilateral can be drawn, if all four sides and one diagonal is known.

SOLUTION

Solution : True
A quadrilateral can be constructed uniquely if four sides and one diagonal is known.

Question 125

Question 125
State whether the statements are True or False.
A quadrilateral can be drawn, when all the four angles and one side is given.

SOLUTION

Solution : False
We cannot draw a unique quadrilateral if four angles and one side is known.

Question 126

Question 126
State whether the statements are True or False.
A quadrilateral can be drawn, if all four sides and one angle is known.

SOLUTION

Solution : True
A quadrilateral can be drawn, if all four sides and one angle is known.

Question 127

Question 127
State whether the statements are True or False.
A quadrilateral can be drawn, if three sides and two diagonals are given.

SOLUTION

Solution : True
A quadrilateral can be drawn, if three sides and two diagonals are given.

Question 128

Question 128
State whether the statements are True or False.
If diagonals of a quadrilateral bisect each other, it must be a parallelogram.

SOLUTION

Solution : True
It is the property of a parallelogram.

Question 129

Question 129
State whether the statements are True or False.
A quadrilateral can be constructed uniquely, if three angles and any two included sides are given.

SOLUTION

Solution : True
We can construct a unique quadrilateral if three angles and two included sides are given.

Question 130

Question 130
State whether the statements are True or False.
A parallelogram can be constructed uniquely, if both diagonals and the angle between them is given.

SOLUTION

Solution : True
We can draw a unique parallelogram, if both diagonals and the angle between them is given.

Question 131

Question 131
State whether the statements are True or False.
A rhombus can be constructed uniquely, if both diagonals are given.

SOLUTION

Solution : True
A rhombus can be constructed uniquely, if both diagonals are given.

Question 132

Question 132
The diagonals of a rhombus are 8 cm and 15 cm. Find its side.

SOLUTION

Solution : Given, AC = 15 cm, BD = 8 cm.

Since, the diagonals of a rhombus bisect each other at $90^{\circ}$ , therefore in the $Δ A O B$ , we have
$A B^{2} = O A^{2} + O B^{2} \Rightarrow A B^{2} = {(\frac{15}{2})}^{2} + {(\frac{8}{2})}^{2} = (7.5)^{2} + (4)^{2} = 56.25 + 16 \Rightarrow A B^{2} = 72.25 \Rightarrow A B = \sqrt{72.25} \Rightarrow A B = 8.5 c m$
Since it is a rhombus, the length of each side is 8.5 cm.

Question 133

Question 133
Two adjacent angles of a parallelogram are in the ratio 1 : 3. Find its angles.

SOLUTION

Solution : Let the adjacent angles of the parallelogram be x and 3x.

Then, we have $x + 3 x = 180^{\circ}$ [ $∵$ adjacent angles of parallelogram are supplementary]
$\Rightarrow 4 x = 180^{\circ} \Rightarrow x = 45^{\circ}$
Thus, the adjacent angles are $45^{\circ}, 135^{\circ}$ .
Hence, the angles are $45^{\circ}, 135^{\circ}, 45^{\circ}, 135^{\circ}$ . [ $∵$ opposite angles in a parallelogram are equal].

Question 134

Question 134
Of the four quadrilaterals - square, rectangle, rhombus and trapezium - one is somewhat different from the others because of its design. Find it and give justification.

SOLUTION

Solution : In square, rectangle and rhombus, opposite sides are parallel and equal. Also, opposite angles are equal. i.e. they all are parallelograms.
But in trapezium, there is only one pair of parallel sides, i.e. it is not a parallelogram.
Therefore, trapezium has a different design.

Question 135

Question 135
In a rectangle ABCD, AB = 25 cm and BC = 15 cm. In what ratio, does the bisector of $∠ C$ divide AB?

SOLUTION

Solution : Given, AB = 25 cm and BC = 15 cm
Now, in rectangle ABCD,

CO is the bisector of $∠ C$ and it divides AB.
$∴ ∠ O C B = ∠ O C D = 45^{\circ}$

In $Δ O C B$ ,
$∠ C B O + ∠ O C B + ∠ C O B = 180^{\circ}$ [angle sum property of triangle]
$90^{\circ} + 45^{\circ} + ∠ C O B = 180^{\circ} ∠ C O B = 180^{\circ} - (90^{\circ} + 45^{\circ}) ∠ C O B = 180^{\circ} - 135^{\circ} = 45^{\circ}$

Now, in $Δ O C B$ ,
$∠ O C B = ∠ C O B$
Then, OB = BC
$\Rightarrow O B = 15 c m$
$\Rightarrow A O = 25 - O B = 25 - 15 = 10 c m$
CO divides AB in the ratio AO : OB
$∴$ CO divides AB in the ratio: $10 : 15 = 2 : 3$

Question 136

Question 136
PQRS is a rectangle. The perpendicular ST from S on PR divides $∠ S$ in the ratio 2 :3. Find $∠ T P Q$ .

SOLUTION

Solution : Given, $S T ⊥ P R$ and ST divides $∠ S$ in the ratio 2 : 3.

So, sum of ratio = 2 + 3 = 5
Now, $∠ T S P = \frac{2}{5} \times 90^{\circ} = 36^{\circ}, ∠ T S R = \frac{3}{5} \times 90^{\circ} = 54^{\circ}$

Also, by the angle sum property of a triangle,
$∠ T P S = 180^{\circ} - (∠ S T P + ∠ T S P) \Rightarrow ∠ T P S = 180^{\circ} - (90^{\circ} + 36^{\circ}) = 54^{\circ}$

We know that, $∠ S P Q = 90^{\circ}$
$\Rightarrow ∠ T P S + ∠ T P Q = 90^{\circ} \Rightarrow 54^{\circ} + ∠ T P Q = 90^{\circ} \Rightarrow ∠ T P Q = 90^{\circ} - 54^{\circ} = 36^{\circ}$

Question 137

Question 137
A photo frame is in the shape of a quadrilateral, with one diagonal longer than the other. Is it a rectangle? Why or why not?

SOLUTION

Solution : No, it cannot be a rectangle because the diagonals of a rectangle are equal.

Question 138

Question 138
The adjacent angles of a parallelogram are $(2 x - 4)^{\circ}$ and $(3 x - 1)^{\circ}$ . Find the measures of all angles of the parallelogram.

SOLUTION

Solution : The adjacent angles of a parallelogram are supplementary.
$∴ (2 x - 4)^{\circ} + (3 x - 1)^{\circ} = 180^{\circ}$
$\Rightarrow 5 x - 5^{\circ} = 180^{\circ} \Rightarrow 5 x = 185^{\circ} \Rightarrow x = \frac{185^{\circ}}{5} \Rightarrow x = 37^{\circ}$

Thus, the adjacent angles are
$2 x - 4 = 2 \times 37^{\circ} - 4 = 74 - 4 = 70^{\circ}$ .
and $3 x - 1 = 3 \times 37^{\circ} - 1 = 111 - 1 = 110^{\circ}$
Hence, the angles are $70^{\circ}, 110^{\circ}, 70^{\circ}, 110^{\circ}$ .
[ $∵$ Opposite angles in a parallelogram are equal].

Question 139

Question 139
The point of intersection of diagonals of a quadrilateral divides one diagonal in the ratio 1 : 2. Can it be a parallelogram? Why or why not?

SOLUTION

Solution : No, it can never be a parallelogram, as the diagonals of a parallelogram divide each other in the ratio 1 : 1.

Question 140

Question 140
The ratio between exterior angle and interior angle of a regular polygon is 1 : 5. Find the number of sides of the polygon.

SOLUTION

Solution : Let the exterior angle and interior angle be x and 5x, respectively.
Then, $x + 5 x = 180^{\circ}$
[ $∵$ exterior angle and corresponding interior angle are supplementary]
$\Rightarrow 6 x = 180^{\circ} \Rightarrow x = \frac{180^{\circ}}{6} \Rightarrow x = E x t e r i o r a n g l e = 30^{\circ}$
$∴$ The number of sides
$= \frac{360^{\circ}}{E x t e r i o r a n g l e}$
$= \frac{360^{\circ}}{30^{\circ}}$
$= 12$

Question 141

Question 141
Two sticks each of length 5 cm are crossing each other such that they bisect each other. What shape is formed by joining their endpoints? Give reason.

SOLUTION

Solution : Sticks can be taken as the diagonals of a quadrilateral.
Now, since they are bisecting each other, therefore the shape formed by joining their end points will be a parallelogram.

Since the sticks are of equal length, the shape formed will be a rectangle.

Question 142

Question 142
Two sticks each of length 7 cm are crossing each other such that they bisect each other at right angles. What shape is formed by joining their end points? Give reason.

SOLUTION

Solution : Sticks can be treated as the diagonals of a quadrilateral.
Now, since the diagonals (sticks) are bisecting each other each other at right angles, therefore the shape formed by joining their end points can be a rhombus or a square.

Both the sticks are of the same length. This means that the diagonals of the quadrilateral formed are equal and also bisect each other perpendicularly.
$∴$ The shape formed is that of a square.

Question 143

Question 143
A playground in the town is in the form of a kite. The perimeter is 106 m. If one of its sides is 23 m, what are the lengths of other three sides?

SOLUTION

Solution : In a kite, two pairs of adjacent sides are equal. Since we know that one side measures 23 m in length, we can assume that the side adjacent to it will also be of the same length.
Let the length of the opposite side be x m. Then, the length of the side adjacent to this side will also be x m.

Perimeter of the playground = $106 m$
$\Rightarrow 23 + 23 + x + x = 106 \Rightarrow 46 + 2 x = 106 \Rightarrow 2 x = 106 - 46 \Rightarrow 2 x = 60 \Rightarrow x = 30 m$
Hence, the lengths of the other three sides are 23m, 30m and 30m.

Question 144

Question 144
In rectangle READ, find $∠ E A R, ∠ R A D$ and $∠ R O D$ .

SOLUTION

Solution : Given, a rectangle READ, in which $∠ R O E = 60^{\circ}$ .
$∴ ∠ R O D = 180^{\circ} - 60^{\circ} = 120^{\circ}$ . [linear pair]
In $Δ E O R,$
$∠ R O E + ∠ D E R + ∠ E R A = 180^{\circ}$ [Angle sum property]
$\Rightarrow 60^{\circ} + ∠ E R A + ∠ E R A = 180^{\circ}$ [ $∵ O E = O R$ , hence $∠ D E R = ∠ E R A$ ]
$\Rightarrow 2 ∠ E R A = 180^{\circ} - 60^{\circ}$
$∠ E R A = \frac{120^{\circ}}{2} = 60^{\circ}$
$\Rightarrow ∠ R A D = 60^{\circ}$ [Alternate angles]

In rectangle READ, $∠ E A D = 90^{\circ} \Rightarrow ∠ E A R + ∠ R A D = 90^{\circ} \Rightarrow ∠ E A R = 90^{\circ} - ∠ R A D = 90^{\circ} - 60^{\circ} = 30^{\circ}$

Question 145

Question 145
In rectangle PAIR, find $∠ A R I, ∠ R M I$ and $∠ P M A$ .

SOLUTION

Solution : Given, $∠ R A I = 35^{\circ}$ .
$∴ ∠ P R A = 35^{\circ}$ . [ $P R ∥ A I$ and AR is transversal]
$\Rightarrow ∠ A R I = 90^{\circ} - ∠ P R A = 90^{\circ} - 35^{\circ} = 55^{\circ}$ .
$∵ R M = I M, ∠ M R I = ∠ M I R = 55^{\circ}$ .

In $Δ R M I,$
$∠ R M I + ∠ M R I + ∠ M I R = 180^{\circ}$ . [Angle sum property]
$\Rightarrow ∠ R M I = 180^{\circ} - 55^{\circ} - 55^{\circ} = 180^{\circ} - 110^{\circ} = 70^{\circ}$
Also, $∠ P M A = ∠ R M I$ [vertically opposite angles]
$\Rightarrow ∠ P M A = 70^{\circ}$

Question 146

Question 146
In parallelogram ABCD, find $∠ B, ∠ C$ and $∠ D$ .

SOLUTION

Solution : In a parallelogram, the opposite angles are equal, therefore $∠ C = ∠ A = 80^{\circ}$ .
Also, adjacent angles are supplementary.
$∴ ∠ A + ∠ B = 180^{\circ} 80^{\circ} + ∠ B = 180^{\circ} ∠ B = 180^{\circ} - 80^{\circ} \Rightarrow ∠ B = 100^{\circ}$
Now, $∠ B = ∠ D$ [Opposite angles are equal]
$∴ ∠ D = 100^{\circ}$ .

Question 147

Question 147
In parallelogram PQRS, O is the mid-point of SQ. Find $∠ S, ∠ R$ , PQ, QR and diagonal PR.

SOLUTION

Solution : Given, $∠ R Q Y = 60^{\circ}$
$∠ S R Q = ∠ R Q Y$   [ $S R ∥ P Y$ , RQ is the transversal]
$\Rightarrow ∠ R = 60^{\circ}$
$∴ ∠ S = 180^{\circ} - ∠ R = 180^{\circ} - 60^{\circ} = 120^{\circ}$     [ $∵$ adjacent angles are supplementary]

Also, SR = 15 cm
$\Rightarrow$ PQ = 15 cm   [ $∵$ opposite sides of a parallelogram are equal]
And PS = 11 cm
$\Rightarrow$ QR = 11 cm   [ $∵$ opposite sides of a parallelogram are equal]
PR is bisected by SQ, so $P R = 2 \times P O = 2 \times 6 = 12 c m$

Question 148

Question 148
In rhombus BEAM, find $∠ A M E$ and $∠ A E M$ .

SOLUTION

Solution : Given, $∠ B A M = 70^{\circ}$ .
We know that, in rhombus, diagonals bisect each other at right angles.
$∴ ∠ B O M = ∠ B O E = ∠ A O M = ∠ A O E = 90^{\circ}$ .
Now, in $Δ A O M$ .
$∠ A O M + ∠ A M O + ∠ O A M = 180^{\circ}$ . [angle sum property of triangle]
$\Rightarrow 90^{\circ} + ∠ A M O + 70^{\circ} = 180^{\circ} \Rightarrow ∠ A M O = 180^{\circ} - 90^{\circ} - 70^{\circ} \Rightarrow ∠ A M O = 20^{\circ}$
Also, AM = BM = BE = EA.
In $Δ A M E$ , we have,
AM = EA
$∴ ∠ A M E = ∠ A E M = 20^{\circ}$ [ $∵$ equal sides make equal angles]

Question 149

Question 149
In parallelogram FIST, find $∠ S F T, ∠ O S T$ and $∠ S T O$ .

SOLUTION

Solution : Given, $∠ F I S = 60^{\circ}$
Now, $∠ F T S = ∠ F I S = 60^{\circ}$ [ $∵$ opposite angles of a parallelogram are equal]
Now, $F T ∥ I S$ and TI is a transversal, therefore $∠ F I O = ∠ S T O = 35^{\circ}$ [alternate angles]

Also, $∠ F O T + ∠ S O T = 180^{\circ}$ [linear pair]
$\Rightarrow 110^{\circ} + ∠ S O T = 180^{\circ}$
$\Rightarrow ∠ S O T = 70^{\circ}$
In $Δ T O S, ∠ T S O + ∠ O T S + ∠ T O S = 180^{\circ}$ [angle sum property of triangle]
$∴ ∠ O S T = 180^{\circ} - (70^{\circ} + 35^{\circ}) = 75^{\circ}$

In $Δ F S T, ∠ S F T + ∠ F T S + ∠ T S F = 180^{\circ}$ [angle sum property of triangle]
$\Rightarrow ∠ S F T = 180^{\circ} - (60^{\circ} + 75^{\circ}) ∴ ∠ S F T = 45^{\circ}$

Question 150

Question 150
In the given parallelogram YOUR, $∠ R U O = 120^{\circ}$ and OY is extended to point S, such that $∠ S R Y = 50^{\circ}$ . Find $∠ Y S R$ .

SOLUTION

Solution : Given, $∠ R U O = 120^{\circ}$ and $∠ S R Y = 50^{\circ}$
$∠ R Y O = ∠ R U O = 120^{\circ}$ [ $∵$ opposite angles of a parallelogram]
Now, $∠ S Y R = 180^{\circ} - ∠ R Y O$ [linear pair]
$= 180^{\circ} - 120^{\circ} = 60^{\circ}$
In $Δ S R Y$ ,
By the angle sum property of a triangle, $∠ S R Y + ∠ R Y S + ∠ Y S R = 180^{\circ}$
$\Rightarrow 50^{\circ} + 60^{\circ} + ∠ Y S R = 180^{\circ} \Rightarrow ∠ Y S R = 180^{\circ} - (50^{\circ} + 60^{\circ}) = 70^{\circ}$ .

Question 151

Question 151
In kite WEAR, $∠ W E A = 70^{\circ}$ and $∠ A R W = 80^{\circ}$ . Find the remaining two angles.

SOLUTION

Solution : Given, in a kite WEAR, $∠ W E A = 70^{\circ}, ∠ A R W = 80^{\circ}$
Now, by the interior angle sum property of a quadrilateral,
$∠ R W E + ∠ W E A + ∠ E A R + ∠ A R W = 360^{\circ} \Rightarrow ∠ R W E + 70^{\circ} + ∠ E A R + 80^{\circ} = 360^{\circ} \Rightarrow ∠ R W E + ∠ E A R = 360^{\circ} - (70^{\circ} + 80^{\circ}) \Rightarrow ∠ R W E + ∠ E A R = 360^{\circ} - 150^{\circ} \Rightarrow ∠ R W E + ∠ E A R = 210^{\circ} \dots \dots (i)$
In $Δ R A W$ , $∠ R W A = ∠ R A W$ $[∵ R W = R A] \dots \dots (i i)$
In $Δ E A W$ , $∠ A W E = ∠ W A E$ $[∵ W E = A E] \dots \dots (i i i)$
On adding Eqs. (ii) and (iii), we get $∠ R W A + ∠ A W E = ∠ R A W + ∠ W A E$
$\Rightarrow ∠ R W E = ∠ R A E$
From Eq. (i),
$2 ∠ R W E = 210^{\circ} ∠ R W E = 105^{\circ} \Rightarrow ∠ R W E = ∠ R A E = 105^{\circ}$

Question 152

Question 152 (i)
A rectangle MORE is shown below.

Answer the following questions by giving an appropriate reason.
Is RE = OM?

SOLUTION

Solution : Yes, RE = OM
Given, MORE is a rectangle. Therefore, opposite sides are equal.

Question 153

Question 152 (ii)
A rectangle MORE is shown below.

Answer the following questions by giving appropriate reason.
Is $∠ M Y O = ∠ R X E$ ?

SOLUTION

Solution : Yes, $∠ M Y O = ∠ R X E$
Here, MY and RX are perpendicular to OE.
Since, $∠ R X O = 90^{\circ} \Rightarrow ∠ R X E = 90^{\circ}$ and $∠ M Y E = 90^{\circ} \Rightarrow ∠ M Y O = 90^{\circ}$ .

Question 154

Question 152 (iii)
A rectangle MORE is shown below.

Answer the following questions by giving an appropriate reason.
Is ∠MOY = ∠REX?

SOLUTION

Solution : Yes, $∠ M O Y = ∠ R E X$
$∵ R E ∥ O M$ and EO is a transversal.
$∴ ∠ M O E = ∠ O E R$ [ $∵$ alternate interior angles]
$\Rightarrow ∠ M O Y = ∠ R E X$ .

Question 155

Question 152 (iv)
A rectangle MORE is shown below.

Answer the following questions by giving an appropriate reason.
Is $Δ M Y O ≅ Δ R X E$ ?

SOLUTION

Solution : Yes, $Δ M Y O ≅ Δ R X E$
In $Δ M Y O$ and $Δ R X E$
MO = RE [opposite sides are equal]
$∠ M O Y = ∠ R E X$         [ $∵$ Alternate angles, $O M ∥ R E$ , OE is the transversal]
$∠ M Y O = ∠ R X E$         [MY and RX are perpendicular to OE]
$∴ Δ M Y O ≅ Δ R X E$      [by AAS]

Question 156

Question 152 (v)
A rectangle MORE is shown below.

Answer the following questions by giving an appropriate reason.
Is MY = RX?

SOLUTION

Solution : Yes, MY = RX
Since these are corresponding parts of congruent triangles.

Question 157

Question 153
In parallelogram LOST, $S N ⊥ O L$ and $S M ⊥ L T$ . Find $∠ S T M, ∠ S O N$ and $∠ N S M$ .

SOLUTION

Solution : Given, $∠ M S T = 40^{\circ}$
In $Δ M S T$ ,
By the angle sum property of a triangle, $∠ T M S + ∠ M S T + ∠ S T M = 180^{\circ}$
$\Rightarrow ∠ S T M = 180^{\circ} - (90^{\circ} + 40^{\circ})$    $[∵ S M ⊥ L T, ∠ T M S = 90^{\circ}]$
$= 50^{\circ}$
$∴ ∠ S O N = ∠ S T M = 50^{\circ}$     [ $∵$ opposite angles of a parallelogram are equal]
Now, in the $Δ O N S$ ,
$∠ O N S + ∠ O S N + ∠ S O N = 180^{\circ}$    [angle sum property of triangle]
$∠ O S N = 180^{\circ} - (90^{\circ} + 50^{\circ})$
$= 180^{\circ} - 140^{\circ} = 40^{\circ}$
Moreover, $∠ S O N + ∠ T S O = 180^{\circ}$
[ $∵$ adjacent angles of a parallelogram are supplementary]
$\Rightarrow ∠ S O N + ∠ T S M + ∠ N S M + ∠ O S N = 180^{\circ} \Rightarrow 50^{\circ} + 40^{\circ} + ∠ N S M + 40^{\circ} = 180^{\circ} \Rightarrow 130^{\circ} + ∠ N S M = 180^{\circ} \Rightarrow ∠ N S M = 180^{\circ} - 130^{\circ} = 50^{\circ}$ .

Question 158

Question 154
In trapezium HARE, EP and RP are bisectors of $∠ E$ and $∠ R$ , respectively. Find $∠ H A R$ and $∠ E H A$ .

SOLUTION

Solution : Given, $∠ P E R = 25^{\circ}$ and $∠ P R E = 30^{\circ}$
Since EP and PR are angle bisectors of $∠ R E H$ and $∠ A R E$ respectively,
$∠ P E H = 25^{\circ}$ and $∠ P R A = 30^{\circ}$

$∠ E + ∠ H = 180^{\circ} a n d ∠ R + ∠ A = 180^{\circ}$ [HARE is a trapezium]

$\Rightarrow ∠ P E R + ∠ P E H + ∠ H = 180^{\circ} a n d$
$∠ E R P + ∠ P R A + ∠ R A H = 180^{\circ}$

$\Rightarrow 25^{\circ} + 25^{\circ} + ∠ H = 180^{\circ} a n d$
$30^{\circ} + 30^{\circ} + ∠ A = 180^{\circ}$

$\Rightarrow$