# Free Exemplar Exercise 3.1 Practice Test - 8th Grade

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

#### SOLUTION

Solution : Given, three angles of a quadrilateral =75
Let the fourth angle be x.
Then, according to the property, 75+75+75+x=360, since sum of the angles of a quadrilateral is 360.
So, 225+x=360 or x=360225=135
Hence, the fourth angle is 135.

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
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.

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
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
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
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
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
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
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
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
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.
Also, an obtuse angle is more than 90 and less than 180.
Thus, all the angles of a quadrilateral cannot be obtuse.
Hence, at the most 3 angles can be obtuse.

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
What is the sum of all the angles of a pentagon?
a) 180
b) 360
c) 540
d) 720

#### SOLUTION

Solution : We know that, the sum of angles of a polygon
=(n2)×180,
where n is the number of sides of the polygon.
In pentagon, n = 5
Sum of the angles
=(n2)×180
=(52)×180
=3×180
=540

Question 15
What is the sum of all angles of a hexagon?
a) 180
b) 360
c) 540
d) 720

#### SOLUTION

Solution : We know that, the sum of angles of a polygon
=(n2)×180,
where n is the number of sides of the polygon.
In hexagon, n = 6
Sum of the angles
=(n2)×180
=(62)×180
=4×180
=720

Question 16
If two adjacent angles of a parallelogram are (5x5) and (10x+35), 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,
(5x5)+(10x+35)=18015x+30=18015x=150x=10
Thus, the angles are (5×105)=45 and (10×10+35)=135.
Hence, the required ratio is 45:135, that is, 1 : 3.

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
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
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
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 :
n(n3)2
In hexagon, n = 6
Number of diagonals in a hexagon =6(63)2=6×32=3×3=9

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
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
The sum of all exterior angles of a triangle is
a) 180
b) 360
c) 540
d) 720

#### SOLUTION

Solution : We know that the sum of all exterior angles of any polygon is 360.
Triangle is also a polygon. Hence, the sum of all exterior angles of a triangle is 360.

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
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.
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
The angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4. The smallest angle is
a) 72
b) 144
c) 36
d) 18

#### 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,2x,3x and 4x.
x+2x+3x+4x=360    [ sum of the angles of a quadrilateral is 360]
10x=360x=36010=36
Hence, the smallest angle is 36.

Question 27
In the trapezium ABCD, the measure of D is
a) 55
b) 115
c) 135
d) 125 #### SOLUTION

Solution : We know that, in a trapezium, the angles on either side of the base are supplementary.
In trapezium ABCD,
A+D=18055+D=180D=18055D=125

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

#### SOLUTION

Solution : Let the fourth angle be x, then 80+80+80+x=360
[ sum of all the angles of quadrilaterals is 360]
240+x=360x=360240x=120

Question 29
The number of sides of a regular polygon in which each exterior angle has a measure of 45 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.
Since each exterior angle measures 45, the number of sides
=Sum of all exterior anglesMeasure of an exterior angle=36045=8

Question 30
In a parallellogram PQRS, if P=60, then other three angles are
a) 45,135,120
b) 60,120,120
c) 60,135,135
d) 45,135,135

#### SOLUTION

Solution : Given, P=60 Since, in a parallelogram, adjacent angles are supplementary,
P+Q=18060+Q=180Q=120
Also, opposite angles are equal in a parallelogram.
Therefore, R=P=60,S=Q=120
Hence, other three angles are 60,120,120.

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

#### SOLUTION

Solution : Given, two adjacent angles of a parallelogram are in the ratio 2 : 3
Let the angles be 2x and 3x.
Then,
2x+3x=180      [ adjacent angles of a parallelogram are supplementary]
5x=180x=36
Hence, the measures of the angles are 2x=2×36=72 and 3x=3×36=108.

Question 32
If PQRS is a parallelogram, then PR is equal to
a) 60
b) 90
c) 80
d) 0

#### SOLUTION

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

Question 33
The sum of adjacent angles of a parallelogram is
a) 180
b) 120
c) 360
d) 90

#### SOLUTION

Solution : By property of the parallelogram, we know that, the sum of adjacent angles of a parallelogram is 180.

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

#### SOLUTION

Solution : Let EC and FC be altitudes and ECF=30. Let EDC=x=FBC
So, ECD=90xandBCF=90x
So, by property of the parallelogram,
ADC+DCB=180ADC+(ECD+ECF+BCF)=180x+90x+30+90x=180x=180210=30x=30
Hence, DCB=30+60+60=150.

Question 35
In the given figure, ABCD and BDCE are parallelograms with common base DC. If BCBD, then BEC is equal to
a) 60
b) 30
c) 150
d) 120 #### SOLUTION

Solution : BAD=30           [given]
BCD=30        [ opposite angles of a parallelogram are equal]
In ΔCBD, by angle sum property of a traingle, we have,
DBC+BCD+CDB=18090+30+CDB=180CDB=180120=60
BEC=CDB=60    [ opposite angles of a parallelogram are equal]

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 ΔBCD,
BDC=90
Using Pythagoras theorem,
BC2=BD2+CD2
BC2=102+242=100+576BC2=676BC=676BC=26 cm

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. If they are equal, then each angle will be of measure :
1802=90.
Hence, such a parallelogram is a rectangle.

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

#### SOLUTION

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

Question 39
The sum of angles of a concave quadrilateral is
a) More than 360
b) Less than 360
c) Equal to 360
d) Twice of 360

#### SOLUTION

Solution : We know that the sum of interior angles of any polygon (convex or concave) having n sides is (n2)×180.
The sum of angles of a concave quadrilateral is (42)×180=360.

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

#### SOLUTION

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

Question 41
In the figure, BEST is a rhombus, then the value of y - x is
a) 40
b) 50
c) 20
d) 10 #### SOLUTION

Solution : Given: BEST is a rhombus, so,
TSBE
and BS is a transversal.
SBE=TSB=40       [alternate interior anges]
Also, y=90              [diagonals of a rhombus  bisect at 90]

In ΔTSO,
STO+TSO+SOT=180   [Angle sum property of a triangle]
x+40+90=180x=1809040=50
yx=9050=40

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
Which of the following is not true for an exterior angle of a regular polygon with n sides?
a) Each exterior angle =360n
b) Exterior angle =180 - Interior angle
c) n=360Exterior angle
d) Each exterior angle =(n2)×180n

#### 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
PQRS is a square. PR and SQ intersect at O. Then, POQ 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,
POQ=90.

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

#### SOLUTION

Solution : Let the adjacent angles of a parallelogram be x and 5x, respectively.
Then, x+5x=180       [ adjacent angles of a parallelogram are supplementary]
6x=180
x=30
The adjacent anlges are 30 and 150.
Hence, the angles are 30,150,30,150 .     [ opposite angles are equal]

Question 46
A parallelogram PQRS is constructed with sides QR = 6 cm, PQ = 4 cm and PQR=90. 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, then all angles will be of 90 and a parallelogram with all angles equal to 90 is called a rectangle.

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 PQRS
c) Trapezium with QRPS
d) Kite

#### SOLUTION

Solution : Let the angles be x, 3x, 7x and 9x, then,
x+3x+7x+9x=360    [ sum of angles in any quadrilateral is 360]
20x=360x=36020x=18
Then, the angles P, Q, R and S are 18,54,126 and 162, respectively.
Since, P+S=18+162=180 and Q+R=54+126=180,
So, PQSR.
The quadrilateral PQRS is a trapezium with PQSR.

Question 48
PQRS is a trapezium in which PQSR and P=130,Q=110. Then, R is equal to
a) 70
b) 50
c) 65
d) 55

#### SOLUTION

Solution : Since, PQRS is a trapezium and PQSR.
Q+R=180    [ angles between the pair of parallel sides are supplementary]
R=180110=70

Question 49
The number of sides of a regular polygon in which each interior angle measures 135 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 360n.

The number of sides,
n=360Exterior angle=360180135

[ exterior angle + interior angle =180]

=36045=8

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
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
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.
Hence, we require only two measurements to construct a unique rectangle.

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
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
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
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
Fill in the blanks to make the statements true.

The sum of all ___ of a quadrilateral is 360.

#### SOLUTION

Solution : Angles
We know that the sum of all angles of a quadrilateral is 360.

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
Measure of exterior angle =360Number of sides=3605=72.        [ in pentagon, number of sides, n = 5]

Question 59
Fill in the blanks to make the statements true.

Sum of the angles of a hexagon is ___.

#### SOLUTION

Solution : 720

The sum of angles of an n-gon (a polygon with n sides)
=(n2)×180.
Sum of the angles of a hexagon =(62)×180=4×180=720.     [ In a hexagon, number of sides, n = 6]

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
We know that, measure of each exterior angle =360Number of sides=36018=20.

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, is ___.

#### SOLUTION

Solution : We know that the sum of exterior angles of a regular polygon is 360.
Given, the measure of each exterior angle = 36

Number of sides
=360Exterior angle

=36036=10.

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.

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
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
=(n2)×180

Measure of each interior angle =Sum of interior anglesNumber of sides=(52)×1805=3×36=108

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
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 =n(n3)2.
Here, n=6, therefore the number of diagonals =6(63)2=6×32=9.

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
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
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 =(n2)×180
=(2n4)×90.

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

The sum of all exterior angles of a polygon is .

#### SOLUTION

Solution : 360

The sum of all exterior angles of a polygon is always 360.

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
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
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
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

___
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
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
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
Fill in the blanks to make the statements true.

The measure of ___ angle of a concave quadrilateral is more than 180.

#### SOLUTION

Solution : One
A concave polygon is a polygon in which at least one interior angle is more than 180.

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
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 is ___.

#### SOLUTION

Solution : 5
We know that,
the sum of exterior angles of any polygon is 360.
The number of sides in a regular polygon =360Exterior angle.
The number of sides in given polygon =36072=5.

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
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)=2×14=28 cm.

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
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
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
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
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
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
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
Fill in the blanks to make the statements true.

In trapezium ABCD with ABCD, if A=100, then D= ___.

#### SOLUTION

Solution : 80 In a trapezium, we know that the angles on either side of the base are supplementary.
So, in trapezium ABCD, given ABCD
A+D=180100+D=180D=180100D=80.

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 and in a quadrilateral, the sum of interior angles is also 360. Therefore, a quadrilateral is a polygon in which the sum of both interior and exterior angles are equal.

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
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
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 but in a kite, the measures of angles can vary.

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
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
State whether the statements are True or False.

Sum of all the angles of a quadrilateral is 180.

#### SOLUTION

Solution : False
The sum of all the angles of a quadrilateral is 360.

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
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 and the sum of exterior angles is 360, which is twice the sum of interior angles.

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
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
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.

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. The sum of interior angles of a hexagon is 720, which is double the sum of exterior angles.

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
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
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
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
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+2x+3x=180           [angle sum property of triangle]
6x=180x=1806=30
The interior angles are 30,60 and 90.
Now, the exterior angles will be (18030),(18060) and (18090)
i.e. 150,120 and 90.
The ratio of exterior angles =150:120:90=15:12:9=5:4:3.

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
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
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
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
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
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
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
State whether the statements are True or False.

Every trapezium is a rectangle.

#### SOLUTION

Solution : False
Since in a rectangle, all angles measure 90, but in a trapezium, the measure of the angles can vary.

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.

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
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
State whether the statements are True or False.

Every square is a trapezium.

#### SOLUTION

Solution : False.
trapezium is a quadrilateral with exactly one pair of parallel sides.
square is a parallelogram with all sides equal and one angle equal to 90.

Question 121
State whether the statements are True or False.

Every rhombus is a trapezium.

#### SOLUTION

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

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
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. This is not possible in case of a quadrilateral.

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
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
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
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
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
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
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
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
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, therefore in the ΔAOB, we  have
AB2=OA2+OB2AB2=(152)2+(82)2=(7.5)2+(4)2=56.25+16AB2=72.25AB=72.25AB=8.5 cm
Since it is a rhombus, the length of each side is 8.5 cm.

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+3x=180    [ adjacent angles of parallelogram are supplementary]
4x=180x=45
Thus, the adjacent angles are 45,135.
Hence, the angles are 45,135,45,135.     [ opposite angles in a parallelogram are equal].

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
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.
OCB=OCD=45

In ΔOCB,
CBO+OCB+COB=180    [angle sum property of triangle]
90+45+COB=180COB=180(90+45)COB=180135=45

Now, in ΔOCB,
OCB=COB
Then, OB = BC
OB=15 cm
AO=25OB=2515=10 cm
CO divides AB in the ratio AO : OB
CO divides AB in the ratio: 10:15=2:3

Question 136
PQRS is a rectangle. The perpendicular ST from S on PR divides S in the ratio 2 :3. Find TPQ.

#### SOLUTION

Solution : Given, STPR and ST divides S in the ratio 2 : 3. So, sum of ratio = 2 + 3 = 5
Now, TSP=25×90=36,TSR=35×90=54

Also, by the angle sum property of a triangle,
TPS=180(STP+TSP)TPS=180(90+36)=54

We know that, SPQ=90
TPS+TPQ=9054+TPQ=90TPQ=9054=36

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
The adjacent angles of a parallelogram are (2x4) and (3x1). Find the measures of all angles of the parallelogram.

#### SOLUTION

Solution : The adjacent angles of a parallelogram are supplementary.
(2x4)+(3x1)=180
5x5=1805x=185x=1855x=37

Thus, the adjacent angles are
2x4=2×374=744=70.
and 3x1=3×371=1111=110
Hence, the angles are 70,110,70,110.
[ Opposite angles in a parallelogram are equal].

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
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+5x=180
[ exterior angle and corresponding interior angle are supplementary]
6x=180x=1806x=Exterior angle=30
The number of sides
=360Exterior angle
=36030
=12

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
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
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
23+23+x+x=10646+2x=1062x=106462x=60x=30 m
Hence, the lengths of the other three sides are 23m, 30m and 30m.

Question 144
In rectangle READ, find EAR,RAD and ROD. #### SOLUTION

Solution : Given, a rectangle READ, in which ROE=60.
ROD=18060=120.      [linear pair]
In ΔEOR,
ROE+DER+ERA=180 [Angle sum property]
60+ERA+ERA=180   [OE=OR, hence  DER=ERA]
2ERA=18060
ERA=1202=60
RAD=60  [Alternate angles]

In rectangle READ, EAD=90EAR+RAD=90EAR=90RAD=9060=30

Question 145
In rectangle PAIR, find ARI,RMI and PMA. #### SOLUTION

Solution : Given, RAI=35.
PRA=35  .   [PRAI and AR is transversal]
ARI=90PRA=9035=55.
RM=IM,MRI=MIR=55.

In ΔRMI,
RMI+MRI+MIR=180 .  [Angle sum property]
RMI=1805555=180110=70
Also, PMA=RMI  [vertically opposite angles]
PMA=70

Question 146
In parallelogram ABCD, find B,C and D. #### SOLUTION

Solution : In a parallelogram, the opposite angles are equal, therefore C=A=80.
Also, adjacent angles are supplementary.
A+B=18080+B=180B=18080B=100
Now, B=D [Opposite angles are equal]
D=100.

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

Solution : Given, RQY=60
SRQ=RQY      [SRPY, RQ is the transversal]
R=60
S=180R=18060=120     [ adjacent angles are supplementary]

Also, SR = 15 cm
PQ = 15 cm   [ opposite sides of a parallelogram are equal]
And PS = 11 cm
QR = 11 cm   [ opposite sides of a parallelogram are equal]
PR is bisected by SQ, so PR=2×PO=2×6=12 cm

Question 148
In rhombus BEAM, find AME and AEM. #### SOLUTION

Solution : Given, BAM=70.
We know that, in rhombus, diagonals bisect each other at right angles.
BOM=BOE=AOM=AOE=90.
Now, in ΔAOM.
AOM+AMO+OAM=180 .   [angle sum property of triangle]
90+AMO+70=180AMO=1809070AMO=20
Also, AM = BM = BE = EA.
In ΔAME, we have,
AM = EA
AME=AEM=20     [ equal sides make equal angles]

Question 149
In parallelogram FIST, find SFT,OST and STO. #### SOLUTION

Solution : Given, FIS=60
Now, FTS=FIS=60    [ opposite angles of a parallelogram are equal]
Now, FTIS and TI is a transversal, therefore FIO=STO=35  [alternate angles]

Also, FOT+SOT=180    [linear pair]
110+SOT=180
SOT=70
In ΔTOS,TSO+OTS+TOS=180   [angle sum property of triangle]
OST=180(70+35)=75

In ΔFST,SFT+FTS+TSF=180  [angle sum property of triangle]
SFT=180(60+75)SFT=45

Question 150
In the given parallelogram YOUR, RUO=120 and OY is extended to point S, such that SRY=50. Find YSR. #### SOLUTION

Solution : Given, RUO=120 and SRY=50
RYO=RUO=120   [ opposite angles of a parallelogram]
Now, SYR=180RYO    [linear pair]
=180120=60
In ΔSRY,
By the angle sum property of a triangle, SRY+RYS+YSR=180
50+60+YSR=180YSR=180(50+60)=70.

Question 151
In kite WEAR, WEA=70 and ARW=80. Find the remaining two angles. #### SOLUTION

Solution : Given, in a kite WEAR, WEA=70,ARW=80
Now, by the interior angle sum property of a quadrilateral,
RWE+WEA+EAR+ARW=360RWE+70+EAR+80=360RWE+EAR=360(70+80)RWE+EAR=360150RWE+EAR=210(i)
In ΔRAW, RWA=RAW      [RW=RA](ii)
In ΔEAW, AWE=WAE      [WE=AE](iii)
On adding Eqs. (ii) and (iii), we get RWA+AWE=RAW+WAE
RWE=RAE
From Eq. (i),
2RWE=210RWE=105RWE=RAE=105

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 152 (ii)
A rectangle MORE is shown below. Answer the following questions by giving appropriate reason.
Is MYO=RXE?

#### SOLUTION

Solution : Yes, MYO=RXE
Here, MY and RX are perpendicular to OE.
Since, RXO=90RXE=90 and MYE=90MYO=90.

Question 152 (iii)
A rectangle MORE is shown below. Answer the following questions by giving an appropriate reason.
Is ∠MOY = ∠REX?

#### SOLUTION

Solution : Yes, MOY=REX
REOM and EO is a transversal.
MOE=OER     [ alternate interior angles]
MOY=REX.

Question 152 (iv)
A rectangle MORE is shown below. Answer the following questions by giving an appropriate reason.
Is ΔMYOΔRXE?

#### SOLUTION

Solution : Yes, ΔMYOΔRXE
In ΔMYO and ΔRXE
MO = RE       [opposite sides are equal]
MOY=REX        [Alternate angles, OMRE, OE is the transversal]
MYO=RXE        [MY and RX are perpendicular to OE]
ΔMYOΔRXE     [by AAS]

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 153
In parallelogram LOST, SNOL and SMLT. Find STM,SON and NSM. #### SOLUTION

Solution : Given, MST=40
In ΔMST,
By the angle sum property of a triangle, TMS+MST+STM=180
STM=180(90+40)    [SMLT,TMS=90]
=50
SON=STM=50     [ opposite angles of a parallelogram are equal]
Now, in the ΔONS,
ONS+OSN+SON=180    [angle sum property of triangle]
OSN=180(90+50)
=180140=40
Moreover, SON+TSO=180
[ adjacent angles of a parallelogram are supplementary]
SON+TSM+NSM+OSN=18050+40+NSM+40=180130+NSM=180NSM=180130=50 .

Question 154
In trapezium HARE, EP and RP are bisectors of E and R, respectively. Find HAR and EHA. #### SOLUTION

Solution : Given, PER=25 and PRE=30
Since EP and PR are angle bisectors of REH and ARE respectively,
PEH=25 and PRA=30

E+H=180 and R+A=180     [HARE is a trapezium]

PER+PEH+H=180 and
ERP+PRA+RAH=180

25+25+H=180 and
30+30+A=180