Math5

Revised Edition

5

Oasis School Mathematics Book-5 1

2 Oasis School Mathematics Book-5

Preface Oasis School Mathematics is an activity-oriented and interactive series
intended for the students of the primary level. It has been designed in
compliance with the latest curriculum of the Curriculum Development
Centre (CDC), the Government of Nepal, with a focus on child psychology
of requiring mathematical knowledge and skill. The major thrust is on
creating an enjoyable experience in learning mathematics through the
inclusion of a variety of problems which are closely related to our daily
life. This book is expected to foster a positive attitude among children and
encourage them to enjoy mathematics. A genuine attempt has been made
to present mathematical concepts with ample illustrations, assignments,
activities, exercises and project works to students so as to encourage them
to participate actively in the process of learning.

Key Features Unit test evaluates
knowledge, understanding
Assignment provides students an and skill of the mathematical
opportunity for immediate practice for concepts learnt in the
retention of mathematical concepts. respective unit.

Illustration facilitates easy groups of logic Worksheet helps students
behind the mathematical concepts. grasp the basic principles of
mathematical concepts in a
Exercise helps students have additional fun way.
practice for reinforcement of mathematical
knowledge and skill. Objective questions help
students analyse the correct
Activity provides opportunities for from multiple choices.
students to relate mathematical concepts
to everyday life.
Project work helps students in learning
by doing.

I am highly grateful to principals, administrators, experts and mathematics teachers
who have provided me with opportunities to anticipate in workshops, teachers' training
programmes and interaction programmes as a resource person. Their invaluable
feedback has helped me immensely to thoroughly revise and comprehensively update
the series.

I am deeply indebted to my students who are -the source of encouragement for me to
go ahead with this project.

I am highly grateful to Mr. Bijay Kumar Basnet for painstakingly editing the content
and the language of the series.

Grateful thanks go to chairman Mr. Sanjaya Kumar Chaudhary and Managing Director
Mr. Harish Chandra Bista for their invaluable support and cooperation in getting this
series published in this shape.

At the end, constructive and practical suggestions of all kinds for further improvement
of the book will be appreciated and incorporated in course of revision.

– Shyam Datta Adhikari

Oasis School Mathematics Book-5 3

1. Angles 5

2. Triangle and Quadrilateral 28

3. Number System 41

4. Four Fundamental Operations 57

5. Properties of whole number 75

6. Fractions and Decimals 100

7. Percentage, Unitary Method, Ratio, Simple 143
Interest, Profit and Loss

8. Measurement of Time and Money 159

9. Measurement of Length, Weight and 174
Capacity

10. Perimeter, Area and Volume 190

11. Chart, Bar Graph, Temperature, Ordered 206
Pairs and Co-ordinates

12. Bills and Budget and Sets 216

13. Algebra 230

14. Model Test Paper 256

4 Oasis School Mathematics Book-5

UNIT

1

12
93

6

Oasis School Mathematics Book-5 5

Angles and their measurement

Vertex and Arms of Angle

Line segments OA and OB meet at O A A
the point O in figure (i), rays OA and Figure. (i) B OB
OB meet at a point O in figure (ii) to
form an angle. Figure. (ii)

Hence, two rays or line segments
with a common end point form an
angle.

The common end point of two rays or
two line segments is called a vertex of
the angle. Two line segments or rays
are called arms of the angle.

Class Assignment

1. Show the angle in the given figure.
a. b.

2. Write the vertex and arms of the given angles.

a. A B b. P

O

O and Q and

Vertex : Vertex :
Arms : Arms :

6 Oasis School Mathematics Book-5

Notation of Angles P

We use capital letters to represent an angle. The OQ
angle is denoted by ‘∠’. So we write angle POQ as Say: Angle POQ or angle QOP
∠POQ. It can also be written as ∠QOP.
write: ∠POQ or ∠QOP
While writing an angle, the vertex should be written
in the middle.

Note: This angle can be simply written as ∠O.

Class Assignment

Name the following angles in both ways.

a. A b. P Q c. D d. O

OE

B O F M N

∠AOB or ∠BOA or or or

Measurement of an Angle
We use a protractor to measure an angle. To measure the angle ABC, follow
the following steps.

A

A

BC 50° C
B

Steps

• Place the protractor in such a way that its centre point O is exactly on the
vertex B.

• Adjust it so that line BC is along 0° - 180° line.

• Start counting from 0°, which is above BC. Note the number of degrees at
the point where the other arm AB passes.

In the above figure, arm AB passes through 50° mark. So ∠ABC = 50°.

Oasis School Mathematics Book-5 7

I have to use inner scale. Its 0° I have to use outer scale. Its
is on the arm OA. scale 0° is on OC.

Remember:
• Use the scale with 0° on one of the arms of the angle.

Class Assignment

a. b. E

A

∠AOB = ∠EOF =

CO BD O F

Acute angle: O A B
An angle whose measurement is less than 90° is called 50°
an acute angle. In the given figure, ∠AOB = 50° which
is less than 90°.

\ ∠AOB is an acute angle.

8 Oasis School Mathematics Book-5

Obtuse angle: A B

An angle whose measurement is more than 90° 110°
and less than 180° is called an obtuse angle. In the O
given figure, ∠AOB = 110°, which is more than 90°
and less than 180°.

\ ∠AOB is an obtuse angle.

Right angle: A
An angle whose measurement is exactly 90° is
called right angle. In the given figure, ∠AOB = 90°. 90°
\ ∠AOB is a right angle. O

Straight angle: A 180° B
O B
An angle whose measurement is 180° is
called straight angle. In the given figure,
∠AOB = 180°.

So, ∠AOB is a straight angle.

Reflex angle: 230° A
O
An angle whose measurement is more than
180° and less than 360° is called a reflex B
angle. ∠AOB shown in the figure is a reflex
angle. Size

Angles Figure Examples

Acute angle less than 30°, 40°, 50°, etc.
90°
Obtuse
angle Acute angle more than 110°, 125°, 145°,
Obtuse angle 90° and etc.
less than
180°

Oasis School Mathematics Book-5 9

Right angle 90° 90°

Straight Right angle 180° 180°
angle Straight angle

Reflex angle more than 225°, 240°, etc.
180° and
Reflex angle less than

Exercise 1.1 360°

a. A b. P c. X

Q

OB R ZY

A E
OF
CO

10 Oasis School Mathematics Book-5

A B P
O OQ

a. A b. D F c. G

BC E H I
Z
d. M e. Q R f. X
Y

O NP

Oasis School Mathematics Book-5 11

8. a. An angle is given in the figure. P

(i) Name the vertex and arms.

(ii) Guess which type of angle is this?

(iii) Measure this angle.

(iv) Which type of angle is this? Q R

(v) Is your guess on the type of angle correct?

b. Take any three points on your copy and draw ∠DEF. Then answer
the following questions.

(i) Name the vertex and arms of this angle.

(ii) Guess which type of angle is this?

(iii) Measure ∠DEF with the help of a protractor.

(iv) Which type of angle is this?

(v) Is your guess correct?

9. Answer the following questions:

a. What is a point called where two arms of an angle meet?

b. What is an angle called whose measurement is exactly 90°?

c. What is an angle called whose measurement is more than 180° and less
than 360°?

d. Which type of angle measures 180°?

e. Is 160° an obtuse angle? Give reason. D

10. In the given figure:

(i) name the angles which are acute A O
(ii) name the angle which is obtuse

(iii) name the angle which is right angle C

(iv) which type of angle is ∠BOC?

B

Answer:

Consult your teacher

12 Oasis School Mathematics Book-5

Construction of angles

With the help of a protractor, we can construct an angle of given measurement.
Look at these examples properly and get an idea about the construction of an
angle.

Example :
Construct ∠ABC = 70°

A

A

B C B C C
A B

BC Steps:

∠ABC is a required angle with • Draw a ray or line segment BC.
measurement of 70°.
• Place the protractor in such a way that its
centre O is exactly on the point B and line BC
along its 0° - 180° line as in the figure.

• Use the scale whose 0° lies on BC.

• Start to count from 0° and mark AB on 70°.

• Remove the protractor and join AB.

Construction of angles 60° and 120° using compass
Draw an angle of 60°, using a compass.

Steps:

• Draw a ray AB

• Take a suitable arc and taking A as the centre

D draw an arc above AB. Mark C where the arc
meets AB.

• With the same arc, taking C as the centre cut at D.

• Join AD.

A C B ∠DAB = 60° is the required angle.

Oasis School Mathematics Book-5 13

Example : Steps:
Draw an angle of 120° using a
compass. • Draw a line AB of suitable length.

F • Take a suitable arc and taking A as the centre
ED draw an arc which meets AB at C.

A CB • Taking C as the centre and using the same arc,
cut the first arc at D.

• Taking D as the centre and using same arc, cut
the first arc again at E.

• Join AE and produce it to F.

∠FAB is the required angle where∠FAB = 120°

Bisection of given angle: Steps:

Example : • With O as centre, draw an arc of suitable
Bisect the given ∠AOB. measure which cuts OA and OB at C and D
respectively.
A
• Taking C and D as centre draw arcs of equal
C radii to intersect at E.
E
• Join OE.
O DB
OE bisects ∠AOB

i.e. ∠AOE = ∠BOE

Exercise 1.2

1. Construct the following angles using a protractor:

a. 15° b. 22° c. 60° d. 65° e. 86° f. 92° g. 105°

h. 112° i. 124° j. 145° k. 160° l. 175° m. 170°

2. Construct the following angles using a compass:

a. 60° b. 120°

3. Construct the following angles with the help of a protractor and bisect them:

a. 80° b. 60° c. 40° d. 110° e. 130°

Answers: Consult your teacher.

14 Oasis School Mathematics Book-5

Pairs of angles

Adjacent angles:

a. B b. A B c. A

AOC O C OB

Fig. (i) Fig. (ii) Fig. (iii) C

In the figures above , all three figures have a I understand ! Adjacent
pair of angles ∠AOB and ∠BOC. angles have common vertex

Both of these angles have common arm OB and a common arm.
and common vertex O.

Therefore, ∠AOB and ∠BOC are a pair of
adjacent angles in each figure.

A pair of angles having a common vertex and a common arm are called adjacent
angles.

If the sum of the adjacent angles is 180°, then they are called a linear pair.
In fig (i), ∠AOB + ∠BOC = 180°. Hence ∠AOB and ∠BOC are the linear pairs.

Class Assignment

In the given figure write the pair of adjacent angles. Their common vertex and a
common arm.

a. A B Adjacent angles : and

Common vertex : .............................

O C Common arm : .............................

b. O Adjacent angles : and
Y Common vertex : .............................
X Z Common arm : .............................

Oasis School Mathematics Book-5 15

c. B Adjacent angles : and

AO Common vertex : .............................

C

Common arm : .............................

Vertically opposite angles: A C
D O
In the given figure, two line segments AB and CD
intersect at point O. The angle ∠AOC and ∠BOD have B
same vertex O but they are on the opposite side of the
vertex O. They are called vertically opposite angles.

Similarly, ∠AOD and ∠BOC are vertically opposite
angles.

Vertically opposite angles are equal. In the figure, ∠AOC = ∠BOD and
∠AOD = ∠BOC.

Class Assignment

Write two pairs of vertically opposite angles in each of the given figures.

a. A D Two pairs of vertically opposite angles are
(i) ............. and .............
(ii) ............. and .............

CB

A
D

Complementary angles
In the given figure, ∠ABC = 90°,
i.e. ∠ABD + ∠CBD = 90°.
These angles are complementary angles. B C
The angles are said to be complementary angles if their sum is 90°. One angle is called
the complement of the other.
Complement of 60° = 90° - 60° = 30°
Complement of 10° = 90° - 10° = 80°

16 Oasis School Mathematics Book-5

Class Assignment

1. Answer the following questions.
a. What is the sum of two complementary angles? ............................

b. Are 30° and 60° complementary angles? ............................

c. Are 20° and 50° complementary angles? ............................

d. In the given figure, ∠AOB and ∠BOC are ............................

2. Fill in the blanks: B C
Complement of 45° =

Complement of 60° = A
Complement of 50° =
65° 25°

O

Complement of x° =

Supplementary angles: A D
In the given figure, ∠CBD is a straight angle,
i.e. ∠CBD = 180° CB
Here, ∠ABD + ∠ABC = ∠CBD I can obtain the supplement
∠ABD + ∠ABC = 180° of an angle by subtracting it
from 180°.
∠ABD and ∠ABC are supplementary angles.
Two angles are said to be supplementary
angles if their sum is 180°. One angle is called
the supplement of the other.
Supplement of 70° = 180° - 70° = 110°
Supplement of 150° = 180° - 150° = 30°

Class Assignment

1. Answer the following questions. C

(a) What is the sum of two supplementary angles?

(b) Are 60° and 120° supplementary angles?

(c) Are 50° and 100° supplementary angles?

(d) In the given figure ∠ADC = 115° and A 115° 65° B
D

∠BDC = 65°.

Oasis School Mathematics Book-5 17

Are ∠ADC and ∠BDC supplementary angles?
2. Fill in the blanks.

Supplement of 80° = Supplement of 120° =

Supplement of 160° = Supplement of x =

Exercise 1.3

1. Copy the given figures in your exercise book and name the pair of
adjacent angles:

a. C b. P S

B

O A O R
D M
c. d. O

FE G

PN

2. Copy the given figures and form the pairs of vertically opposite angles
by colouring with same colour.

a. A C b. E G c. P
O
E S R

O
Q

D B HF

3. From the given figure, write the vertically opposite angle of the angle

given:

a. A C

Vertically opposite angle of ∠AOD = ..........

Vertically opposite angle of ∠AOC = ..........

O

DB

18 Oasis School Mathematics Book-5

b. In the given figure, P S
R O
(i) Which is the vertically opposite angle of
∠POS Q

(ii) Which is the vertically opposite angle of
∠SOQ

(iii) If ∠POS = 70°, then find ∠QOR

(iv) If ∠SOQ = 110°, then find ∠POR

4. Identify which of the following figures have complementary angles:

a. A b. P c. C

R D

B

O QO E
C

5. Identify which of the following figures have supplementary angles:

a. A B C b. P c. C

F

RQ SD E
O
e. 75°
6. Find the complement of the given angles:

a. 42° b. 40° c. 35° d. 65°

f. x° g. y° h. z°

7. Find the supplement of the given angles:

a. 100° b. 70° c. 85° d. 115° e. 125°

f. x° g. y° h. z°

8. a. If a and 52° are vertically opposite angles, find the value of ‘a’.
b. If x and 45° are the linear pairs, find the value of x.
c. If y and 115° are supplementary angles, find the value of y.
d. If z and 63° are complementary angles, find the value of z.

Oasis School Mathematics Book-5 19

9. Find the values of ‘a’ and ‘b’ from the given figures.

a. C b. D

B

50° A 130° C
Oa a

c. AB

b d.
30° a
a
70°

130° b

10. Find the value of each angle from the given figures:

a. b. c.

a 25° 4x 2x
b 5x 4x

d. e. f.

3x x 2x
6x 60°
y 45°
z x

11. From the given two figures, measure ∠AOC, ∠BOC, ∠AOD and ∠BOD
and draw the conclusion:

a. C ∠AOC = ............

A

∠BOD = ............

O ∠BOC = ............

∠AOD = ............

DB

20 Oasis School Mathematics Book-5

b. A C ∠AOC = ............

∠BOD = ............
O ∠BOC = ............

D B ∠AOD = ............

Answers : Consult your teacher.

Parallel lines

Lines which do not meet however far they are A E B
extended are called parallel lines. C G D

In the given figure, AB and CD are parallel lines. A H B
C D
In the given figure, AB and CD are two parallel F
line segments. A line EF meets AB and CD at G B
and H respectively. E D
ab
The line EF is the transversal. B
G D
Angles made by transversal with parallel lines.
H
Exterior angles: A cd
C F
In the given figure, AB and CD are the parallel
lines and EF is the transversal. E
G
The angles a, b, c and d are outside the parallel ab
lines AB and CD. Hence, a, b, c and d are exterior cd
angles. H
F
Interior angles: A
C
In the given figure, a, b, c and d are inside the
parallel lines AB and CD.

The angles a, b, c and d are interior angles.

Oasis School Mathematics Book-5 21

Co - interior angles: E B
G D
In the given figure, AB and CD are the parallel lines. ab
The angles ‘a’ and ‘c’ are the interior angles lying A B
on the same side of transversal. They are co-interior cd D
angles. Similarly, b and d are the co-interior angles. C H
B
Sum of co-interior angles is 180°. F D

a + c = 180° and b + d = 180°

Alternate angles:

In the given figure, AB is parallel to CD. ‘a’ and ‘d’ E
are the interior angles lying on the opposite side A G
ab
of the transversal and they do not have the same
cd
vertex. C H

Therefore ‘a’ and ‘d’ are alternate angles F

Similarly ‘b’ and ‘c’ are alternate angles.

A pair of alternate angles are equal

∠a = ∠d and ∠b = ∠c

Corresponding angles: E
ba
In the given figure, AB is parallel to CD. b and A cd
f lie on the same side of the transversal and one
fe
is interior and the other is exterior angle. b and f hg

are corresponding angles. C F
Similarly, a and e are corresponding angles,

d and g are corresponding angles,
c and h are corresponding angles.
Corresponding angles are equal.
Here, b = f, d = g, a = e and c = h

Class Assignment

1. Locate the co-interior angle of the given angle:

a. b. c.

22 Oasis School Mathematics Book-5

2. Locate the alternate angle of the given angles: c.
a. b.

3. Draw the given figures and locate the corresponding angle of the
given angle:

a. b. c.

Remember !

• Sum of co-interior is 180°
a + b = 180°

• Alternate angles are equal.
a=b

• Corresponding angles are equal.
a=b

Exercise 1.4

1. Study the given figure properly and answer the questions given below.
a. Which is the alternate angle of d?
b. Which is the alternate angle of c?
c. Which two pairs of angles are co-interior angles?
d. Which is the corresponding angle of a?
e. Which is the corresponding angle of b?
f. Which is the corresponding angle of d?
g. Which four pairs of angles are vertically opposite angles?

Oasis School Mathematics Book-5 23

2. a. If a and 50° are the alternate angles, find the value of a.
b. If b and 50° are the corresponding angles, find the value of b.
c. If x and 85° are the co-interior angles, find the value of x.
d. (2x + 5)° and 75° are the alternate angles. Find the value of x.
e. 2y and 120° are the corresponding angles. Find the value of y.
f. (x + 15)° and 75° are the co-interior angles. Find the value of x.

3. From the given figures, find the value of unknown angles.

Answers: Consult your teacher.
24 Oasis School Mathematics Book-5

Worksheet

Observe the given letters and find the number of different types of angles in it.

Letters Number of Number of Number of Number
acute angle obtuse angle right angle of straight

angle

Oasis School Mathematics Book-5 25

Objective Questions

Colour the correct alternatives.
1. In an angle, meeting point of two line segments or two rays is called

arm vertex angle

2. Value of an obtuse angle lies between

0° and 90° 90° and 180° 180° and 360°

3. Which of the following statements is not true?

A reflex angle is greater than An acute angle is A straight angle
180° and less the 360° less than 90° is 90°

4. An angle which is greater than 180° and less than 360° is

acute angle obtuse angle reflex angle

5. In the given figure, vertically opposite angle of ∠AOC is A D

∠BOD ∠AOD ∠BOC O

CB

6. If the sum of two adjacent angles is 180°, then the angles are called

vertically opposite linear pair complementary angles
angles

7. Complement of 60° is

60° 120° 30°

8. Supplement of 80° is 10° 80°

100°

9 If 2x and 60° are alternate angles, then the value of x is

60° 120° 30°

10. In the given figure, co-interior angle of b is d ab
c a° cd

Number of correct answers

26 Oasis School Mathematics Book-5

Unit Test

Full marks: 20

1. Without measuring angles, identify the types of the given angles. 2

a. A b.

OB X OY

c. A d. P

BO Q

2. Find the complement of the given angles. 2
2
a. 42° b. 63°
Q
3. Find the supplement of the given angles.

a. 78° b. y°

P

4. Measure the ∠POQ with the help of a protractor.

O

5. Find the value of unknown angles. 2x3=6

a. b. c. x 60°

60° 2x a yz
3x b c 40° ab
dc

6. Answer the following questions. 4x1=4

a. What is the value of a straight angle?

b. What is a pair of adjacent angles called if their sum is 180°?

c. What is a point called where two arms of an angle meet?

7. a. Construct an angle 80° with the help of a protractor. 2
b. Construct an angle 60° with the help of a compass. 2

Oasis School Mathematics Book-5 27

UNIT Triangle and
Quadrilateral
2

12 Estimated Teaching Hours: 15
93

6

Contents • Types of triangle on the basis of its sides
and angles

• Sum of the angles of a triangle
• Sum of the angles of a quadrilateral

Expected Learning Outcomes

Upon completion of the unit, students will be
able to develop the following competencies:

• To classify the triangles on the basis of its sides
• To classify the triangles on the basis of angles
• To use the property of a triangle that sum of its angles is

180°
• To use the property of a quadrilateral that sum of its

angles is 360°

Materials Required : Models of different triangles, models of different
quadrilaterals, protector, set square, compass, etc.

28 Oasis School Mathematics Book-5

Triangle and Quadrilateral

Triangle:

A triangle is a closed figure bounded by three line A
segments.

In the given figure, three line segments AB, BC and
AC form a triangle ABC.

The line segments are called the sides of the B C
triangle. The meeting points of the sides are called
the vertices of the triangle.

\ AB, BC and AC are the sides of triangle
ABC.

A, B and C are the vertices of triangle ABC. Triangle ABC is denoted
∠A, ∠B and ∠C are the angles of triangle ABC. by DABC.
‘D’ is a symbol to denote a triangle.

P Figure : DPQR
QR Sides : PQ, QR and PR
Vertices : P, Q and R
Angles : ∠P, ∠Q and ∠R

Class Assignment

1. Write the name of given triangle, its sides, its angles and its vertices.

Name of the triangle is ............................. P R
Q
Its sides are ......................, ......................
and ......................

Its angles are ..................., ................... and
...................

Its vertices are ..................., ................... and
...................

Oasis School Mathematics Book-5 29

Name of the triangle is ................... X
Its sides are ................... , ................... and ...................
It angles are ................... , ................... and ................... Y Z
Its vertices are ..................., ................... and ...................

Classification of triangle
A triangle is classified on the basis of its sides and angles.
Types of triangles on the basis of sides:

i. Scalene triangle: 4 cm 4 cm 3 cmA
5 cm
In the given figure, AB = 3 cm, BC = 6 cm and AC = 5 cm.
B 6 cm C
None of the sides are equal, so this type of triangle is
scalene triangle.

If all three sides of a triangle are not equal in length,
such triangle is called a scalene triangle.

ii. Isosceles triangle:

In the given figure, PQ = 4 cm, PR = 4 cm and QR = 5 cm P 4 cm 4 cm
PQ = PR 5 cm

Two sides out of three are equal.

This type of triangle is isosceles triangle. Q R

If any two sides of a triangle are equal, such triangle is

called an isosceles triangle.

iii. Equilateral triangle: X

In the given figure, XY = 4 cm, YZ = 4 cm XZ = 4 cm.
All three sides are equal. This type of triangle is equilateral
triangle.

If all three sides of a triangle are equal, such triangle is Y 4 cm Z
called an equilateral triangle.

Scalene triangle none of the sides are equal
Isosceles triangle any two sides are equal
Equilateral triangle all three sides are equal

30 Oasis School Mathematics Book-5

Class Assignment

1. Answer the following questions:

a. What is a triangle called if all its sides are unequal?

................................

b. In which type of triangle, any two sides are equal in length?

..................................

c. What type of triangle is called an equilateral triangle?

...................................

2. Classify the given triangles on the basis of their sides (scalene, isosceles
or equilateral):

a. A b. P c. A

6 cm
3.5 cm
5 cm
5 cm

4 cm
4.5 cm

B 4.8 cm C Q 3.5 cm R B 4 cm C

Types of triangles on the basis of angles A
Acute angled triangle:
C
In triangle ABC, we can see that all three angles are acute.
Such a triangle is acute angled triangle. R
If all three angles of a triangle are acute, such a triangle is B
called acute angled triangle.

Remember !
Each angle of an acute angled triangle is acute.

Obtuse angled triangle: P
In DPQR, ∠P and ∠R are acute and ∠Q is obtuse.

DPQR is an obtuse angled triangle.

If any one angle of a triangle is obtuse, such a triangle is Q
called obtuse angled triangle.

Oasis School Mathematics Book-5 31

Remember !
An angle of an obtuse angled triangle is obtuse and other two are acute.

Right angled triangle: D F
E
In DDEF, ∠E = 90°. DDEF is a right angled triangle.

If an angle of a triangle is 90°, such a triangle is
called a right angled triangle. Two angles of a right
angled triangle are acute.

Remember !
An angle of a right angled triangle is 90° and remaining two are acute.

Acute angled triangle all three angles are acute
Obtuse angled triangle any one angle is obtuse
Right angled triangle any one angle is right angle

Class Assignment

1. Answer the following questions.

(a) What is a triangle called whose all angles are acute? ......................

(b) What is a triangle called whose one angle is obtuse and two angles
are acute? ......................

(c) What is a triangle called whose one angle is a right angle and two
angles are acute? ......................

2. Classify the following triangles on the basis of angles.

(a) (b)

650 500

550 600

400

(c) 200
150
1450

32 Oasis School Mathematics Book-5

The sum of the angles of a triangle: P

A

Q R
BC

Let's measure the angles of DABC and DPQR

∠A = 45° Again,

∠B = 75° ∠P = 95°

∠C = 60° ∠Q = 50°

What is their sum? ∠R = 35°

∠A + ∠B + ∠C What is their sum?

= 45° + 75° + 60° ∠P +∠Q + ∠R = 180°

= 180°

In both triangles, sum of three angles is 180°.

Hence, the sum of three angles of a triangle is 180°.

Class Assignment A

Measure all three angles of DABC and
verify that the sum of three angles of
a triangle is 180°.

B C

Conclusion: ...........................................................................

Exercise 2.1

1. Classify the given triangles on the basis of their sides (scalene, isosceles

or equilateral):

a. E b. c. A

M
3.5 cm
5 cm
5 cm
5 cm
7 cm 4 cm

D 5 cm F O 4 cm N C
B 2.5 cm

Oasis School Mathematics Book-5 33

2. Classify the triangles on the basis of the given measurement of
sides:

a. 4cm, 7 cm, 6cm b. 3.5 cm, 4 cm, 5cm

c. 4.5 cm, 5 cm, 4.5 cm d. 4.5cm, 5 cm, 5.5cm

e. 6cm, 4cm, 5cm f. 5.5 cm, 6 cm, 5.5cm g. 4 cm, 4cm, 4cm

3. Measure all three sides of DABC and identify the type of triangle on

the basis of sides:
a. A

AB AC BC
..........cm ..........cm ..........cm

B A Type of triangle: .......................

b. C

AB AC BC
..........cm ..........cm ..........cm

B C Type of triangle: .......................

4. Classify the given triangles on the basis of their angles. (acute angled,
obtuse angled and right angled)

a. b. c.

60° 50° 15°

70° 50° 90° 40° 25° 140°

5. Classify the triangles on the basis of the given measurement of
angles:

a. 45°, 65°, 70° b. 60°, 50°, 70° c. 105°, 35°, 40°

d. 90°, 60°, 30° e. 40°, 50°, 80° f. 115°, 45°, 20°

6. Measure each angle of the given triangles and identify whether they
are acute angled, right angled or obtuse angled triangle.

a. A b. D c. P

B CE FQ R

34 Oasis School Mathematics Book-5

7. Find the value of unknown angles from the given figures and identify
the type of triangles on the basis of their angles:

a. b. c.

59° 60° 20°

a 51° 70° x 120° y°
30° x
d. e. f.

b

25° z 30° 45°
85°
g. h. i.
y
30° 25° 125°

15°

80° a

8. a. If three angles of a triangle are 60°, 70° and x°, find the value of x.
b. If three angles of a triangle are (x - 10°), (x + 10°) and x°, find the

value of each angle.
c. If three angles of a triangle are x, 2x and 3x. Find the value of each

angle.

Answers: Consult your teacher.

Quadrilateral A

A quadrilateral is a closed figure bounded by B

four line segments. In the given figure, four

line segments AB, BC, CD and AD form a D

quadrilateral ABCD. C

The line segments are called the sides of quadrilateral. The meeting points of

the sides of the quadrilateral are called the vertices of quadrilateral.

AB, BC, CD and AD are the sides of quadrilateral ABCD.

A, B, C and D are the vertices of quadrilateral ABCD.

∠A, ∠B, ∠C and∠D are the angles of quadrilateral ABCD.

Oasis School Mathematics Book-5 35

The sum of the angles of a quadrilateral

Let's take a quadrilateral ABCD and measure each angle.

∠A = 110° A B
C
∠B = 105°

∠C = 65°

∠D = 80° D

Here, ∠A + ∠B + ∠C +∠D = 110° + 105° + 65° + 80°

= 360°

Hence the sum of four angles of a quadrilateral is 360°.

Class Assignment

Draw a quadrilateral PQRS. Measure all four angles and verify that the sum of
four angles of quadrilateral is 360°.

P

S
Q

R

Conclusion: ...........................................................................

Exercise 2.2

1. Copy the given figure in your exercise book and name the angles, sides

and vertices.

a. X b. p

q

WY s

Z
r

36 Oasis School Mathematics Book-5

2. Measure each of the angles of the given quadrilaterals with the help of
a protractor and also find their sum:

a. A b.

B P Q

D C S R
N F
c. d. E

M

HG
P

O

3. Find the values of unknown angles of the given quadrilaterals:

a. A B b. E
130°
120° F 95° 100°

D 60° xC GY 75°
H
c. P Q d. W
50° Z
130° x 140°

Sx 115° 60° Y
R 45° D
X
e 110° N
f. A
M x
115° O 4x
B 105°
P 70°
3x
C

Oasis School Mathematics Book-5 37

g. E h. Q
x 80°
D M 100°
5x PR

G 90° 3x N 85° xO

i. F j. P

A E yS
110° 120° 150°

yB

Dx C 60° 110°
Q RT

4. a. What is the sum of four angles of a quadrilateral?

b. If four angles of a quadrilatreral are 25°, 75°, 105° and a, find the value of ‘a’.

c. If 2x°, (x - 40)°, 50° and 110° are four angles of a quadrilateral, find the
value of each angle.

d. If 2x°, (x + 20°), (2x - 10°) and 3x° are four angles of a quadrilateral,
find the value of each angle in degrees.

5. a. In a quadrilateral ABCD, ∠A = 70°, ∠B = 80°, ∠C = 100°, find ∠D.
b. In a quadrilateral PQRS, ∠P = 90°, ∠Q = 80°, ∠R = 100°, find the value of ∠S.

Answers: Consult your teacher.

Activity

Sum of the angles of a triangle.
Draw a triangle ABC as given in the figure.

A

BC

figure (i) figure (ii)

- Cut out these as in fig (i).

- Bring A, B and C at one point as shown in figure (ii).

- It will be seen that three angles form a straight line.

Hence, the sum of three angles of a triangle is 180°.

38 Oasis School Mathematics Book-5

Objective Questions

Colour the correct alternatives
1. I am a triangle. All my sides are equal in length. Who am I?

Scalene triangle Isosceles triangle Equilateral triangle

2. I am a triangle. One of my angles is 90° and remaining two angles are acute.
Who am I?

acute angled right angled obtuse angled
triangle triangle triangle

3. How many angles of an obtuse angled triangle are acute?

two one three

4. The sum of three angles of a triangle is equal to 90°

180° 360°

5. Three angles of a triangle are 140°, 25° and 15°. The third angle is

acute angled obtuse angled right angled
triangle triangle triangle

6. Two angles of a triangle are 65° and 45°. Find the third angle.

70° 60° 50°

7. Which of the following statement is true?

All angles of a right All angles of an acute
angled triangle are 90°. angled triangle are acute.

All angles of an obtuse
angled triangle are obtuse.

8. Three angles of a quadrilateral are 20°, 80° and 110°. The fourth angle is

130° 150° 110°

Number of correct answers

Oasis School Mathematics Book-5 39

Unit Test Full marks: 15

1. (a) Classify the given triangle ABC on A 5.7cm
the basis of its sides. 4cm
1
B 6.2cm
C

P 1

(b) Classify the given triangle PQR on

the basis of its angles. 30°

(c) What is a triangle called whose one Q 115° 35° 1
angle is 90°?
R

2. (a) Find the value of x in the given figure. 3×2=6

X b. M xP c. P T
120°
a. 95° 130°
yS

x 25° Z 45° 140° O Q xR
Y N

3. Solve the following questions. 2×2=4

a. Three angles of a triangle are (x - 10)°, x° and (x + 10)°. Find the
value of each angle.

b. If 35°, 85° and 115° are any three angles of a quadrilateral, find its
fourth angle.

4. Draw a DABC, measure ∠A, ∠B and ∠C. Prove by measurement that

∠A + ∠B + ∠C = 180°. 2

40 Oasis School Mathematics Book-5

UNIT

3 Number System

12 Estimated Teaching Hours: 15
93

6

Contents • Place value of the digits of a number
• Nepali and international place value system
• Expanded and standard form of a number

Expected Learning Outcomes

Upon completion of the unit, students will be
able to develop the following competencies:

• To read and write the number and number names using both
Nepali and International place value system

• To use comma in both place value system
• To write the number in expanded and standard form
• To form the numbers from the given digits

Materials Required : Place value chart, flash card of numbers, etc.
Oasis School Mathematics Book-5 41

Hindu - Arabic number system

In about 100 A.D., the Hindus invented the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
The Arabs learnt it from the Hindus and spread the system all over the world.
So the system is called Hindu Arabic number system.
With these ten digits we can write any large number.

Place value of the digits of a number

In the previous class, we have already learnt about the place value of the
numbers up to 7 digits.
A place value chart is used for writing the numbers which have more than two
digits. The given table shows the place value chart for larger numbers.

Period Period Period Period Period
Arabs Crores Lakhs Thousands Ones

Ten Arabs
(10000000000)

Arabs
(1000000000)
Ten Crores
(100000000)

Crores
(10000000)
Ten Lakhs
(1000000)

Lakhs
(100000)
Ten Thousands

(10000)

Thousands
(1000)

Hundreds
(100)
Tens
(10)
Ones
(1)

Arabs, crores, lakhs, thousands and ones are the periods. Ones, tens, thousands,
ten thousands, lakhs, ten lakhs, crores, ten crores, arabs, ten arabs are the
places of the digits in a number.
Place value of a digit = digit × its place
Let's take a number 46281794613 and discuss the place value of its digits.

42 Oasis School Mathematics Book-5

4 6 2 8 1 7 9 4 6 1 3 Place Number × its place Place value

Ones 3 × 1 3
10
Tens 1 × 10 600
4000
Hundreds 6 × 100 90000
700000
Thousands 4 × 1000 1000000
80000000
Ten thousands 9 × 10000 200000000
6000000000
Lakhs 7 × 100000 40000000000

Ten lakhs 1 × 1000000 Ones

Crores 8 × 10000000

Ten crores 2 × 100000000

Arabs 6 × 1000000000

Ten arabs 4 × 10000000000

Showing this number in place value chart

Arabs Crores Lakhs Thousands

Ten Arabs
Arabs

Ten Crores
Crores

Ten Lakhs
Lakhs

Ten Thousands

Thousands
Hundred

Tens
Ones

4628179461 3

The number name is: While reading the number, we
have to read the number, along
Forty six arab twenty eight crore
seventeen lakh ninety four thousand with their periods.
six hundred thirteen.

Oasis School Mathematics Book-5 43

Types of place value system

There are two types of place value system in which the numbers are arranged.
They are :

a. Nepali place value system b. International place value system

Nepali place value system
Ones, tens, hundreds, thousands, ten thousands, etc are the places of Nepali
place value system and ones, thousands, lakhs, crores etc are the periods.

Period Place Place values

Ones Ones 1

Tens 10

Hundreds 100

Thousands Thousands 1000

Ten thousands 10000

Lakhs Lakhs 100000

Ten lakhs 1000000

Crores Crores 10000000

Ten crores 100000000

Arabs Arabs 1000000000

Ten arabs 10000000000

Let's write a number in place value chart according to Nepali system of
numeration.
Take a number 543216159.

Period Crores Lakhs Thousands Ones

Place Ten
crores
Crores

Ten
lakhs
Lakhs

Ten
thousands
thousands

Hundreds
Tens
Ones

54321 6 159

The name of this number is:
Fifty four crore thirty two lakh sixteen thousand one hundred fifty nine.
Use of comma in Nepali place value system
Comma separates the periods of the number, so there are different methods of
using comma in Nepali and international place value system.
74,26,83,91,214

44 Oasis School Mathematics Book-5

Starting from the right, put the first Separate the periods ones, thousands,
comma after three digits then put lakhs, crores and arabs by using comma.
commas in the gap of two digits.
74,26,83,91,214

Class Assignment

1. Show the number 274821093 in the place value chart and write its number
name.

2. Separate the periods using comma according to Nepali place value
system and write their number names.
56041292
Number name: ..................................................................................................
..............................................................................................................................

508264316
Number name: .................................................................................................
.............................................................................................................................

432045932
Number name: .................................................................................................
.............................................................................................................................

3. Write the place value of each of the digits of the given number.

46215328

Oasis School Mathematics Book-5 45

International place value system

Ones, tens, hundreds, thousands, ten thousands, hundred thousands, millions
etc are the places of international place value system of numeration. Ones,
thousands, millions, billions, etc. are its periods.

Period Place Place values
Ones Ones 1
Tens 10
Thousands Hundreds 100
Thousands 1000
Millions Ten thousands 10000
Hundred thousands 100000
Billions Millions 1000000
Ten millions 10000000
Hundred millions 100000000
Billions 1000000000
Ten billions 10000000000
Hundred billions 100000000000

Let's take a number 24384176427 and write it in a place value chart according
to international system of numeration.

Billions Millions Thousands Ones

Ten
billions
Billions
Hundred
millions

Ten
millions
Millions
Hundred
thousands

Ten
thousands
Thousands
Hundreds

Tens
Ones

24384176427

The name of this number is,
Twenty four billions three hundred eighty four millions one hundred seventy
six thousand four hundred and twenty seven.

Comparison of Nepali and International place value system
Let's write a number in Nepali as well as International place value chart and
obtain the relation of their place value.
Take a number 34729437214

46 Oasis School Mathematics Book-5

Kharabs Billions Arabs Ten crores Millions Ten lakhs Lakhs Ten Ten Thousands Thousands Hundreds Hundreds Tens Ones
Ten arabs Crores thousands thousands

Hundred Hundred Hundred Tens Ones
billions millions thousands

Ten Ten
billions millions

34729437214

According to Nepali place value system, the number name is:
Thirty four arab seventy two crore ninety four lakh thirty seven thousand two
hundred fourteen.
According to international system, the number name is:

Thirty four billion seven hundred twenty nine million four hundred thirty
seven thousand two hundred and fourteen.

1 lakh = 100 thousand 1 arab = 1 billion
10 lakhs = 1 million 10 arabs = 10 billion
1 crore = 10 million 1 kharab = 100 billion
10 crores = 100 million

Use of comma in international place value system
In case of international place value system, 43,270,146,945.

Starting from the right put the Separate the periods ones, thousands,
comma in the gap of three digits. millions, billions by using comma.

43,270,146,945.

Class Assignment

1. What are the periods in International place value system?
............................................................................................................................
............................................................................................................................

Oasis School Mathematics Book-5 47

2. Show the number 238204152 in the place value chart and write its
number name using international place value system.

Number name: ..................................................................................................

3. Separate the period using comma and write the number name using
international place value system.
450321482
Number name: ....................................................................................................
824316096
Number name: ...................................................................................................
571862159
Number name: ....................................................................................................

Exercise 3.1

1. Write the following numbers in place value chart according to Nepali as well
as international system of numeration and also write their number names:

a. 132706432 b. 26147902564 c. 54326901427

d. 5732641352 e. 2503261451 f. 28415732451

2. Put the commas in each number according to Nepali as well as
international place value system and write their number names:

a. 437893216 b. 5021037418 c. 4318649357

d. 63740189425 e. 76201050112

3. Write the numerals for the given number name, using comma.

a. Forty two crore sixty three lakh seventy eight thousand five hundred
twenty seven

b. Sixty crore eighty four lakh eighty seven thousand seven hundred
forty three

c. Twenty one arab seventy nine crore sixty two thousand eight
hundred twenty one

d. Seventy nine arab thirty two crore eight lakh forty nine thousand
two hundred twenty seven

e. Two kharab, fourteen arab thirty two crore seven lakh seventeen
thousand six hundred twenty eight

48 Oasis School Mathematics Book-5

4. Show the following numbers in the place value chart according to
international system of numeration and write their number name.

a. 2384096342 b. 5639842014 c. 500215026

d. 1932645357 e. 2670807365

5. Put comma (,) in the following numbers using international place value
system of numeration and write their name.

a. 459304321 b. 703841932 c. 5670863214

d. 234316142 e. 502100214

6. Write the numerals for the given number name, using comma:
a. Thirty seven million, two hundred forty six thousand eight hundred
thirty nine

b. Two hundred twenty six million, five hundred sixty nine thousand
eight hundred sixty five

c. Three billion five hundred twenty one million sixty three thousand
seven hundred eighty one

d. Forty six billion two hundred eighty eight million three hundred
forty six thousand nine hundred eighty one

e. Five hundred eighty six billion twenty five million eight hundred
thirty one thousand nine hundred sixty four

7. Write the place value of underlined digits:

a. 3685092634 b. 9265820193 c. 87231409143
f. 12374281314
d. 5724390123 e. 5792163743

8. Write the place value of each of the digits of the number
25013250631:

9. Answer the following questions:

a. How many lakhs are there in 1 million?
b. How many thousands are there in 1 lakh?
c. How many millions are there in 1 crore?
d. How many crores are there in 100 million?
e. How many billions are there in 1 arab?

Oasis School Mathematics Book-5 49

Expanded and standard form of a number

Place value of any digit of a number is the Place value = digit × its place.
product of the digit and its place. So the
expanded form of a number is the sum of all Expanded form of a number = sum
the place values of the digits. of the place value of all digits.

Let's write the number 2463210897 in expanded form.

Arabs
(1000000000)
Ten Crores
(100000000)

Crores
(10000000)
Ten Lakhs
(1000000)

Lakhs
(100000)
Ten Thousands
(10000)
Thousands
(1000)
Hundred

(100)
Tens
(10)
Ones
(1)

2463210897

\ 2463210897 = 2 × 1000000000 + 4 × 100000000 + 6 × 10000000 + 3 × 1000000 + 2 ×
100000 + 1 × 10000 + 0 × 1000 + 8 × 100 + 9 × 10 + 7 × 1

Standard form Expanded form

Class Assignment

1. Write the given numbers in expanded form.
a. 75640218 = .................. + .................. + .................. + .................. +

.................. + .................. + .................. + ..................
b. 462839702 = .................. + .................. + .................. + .................. +

.................. + .................. + .................. + .................. + ..................

2. Write in standard form:
a. 4 × 100000000 + 3 × 100000000 + 6 × 100000000 + 0 × 10000000 + 4 ×
100000 + 5 × 10000 + 0 × 1000 + 8 × 100 + 7 × 10 + 6 = .............................
a. 6 × 100000000 + 4 × 1000000 + 2 × 100000 + 3 × 10000 + 0 × 1000 + 2 ×
100 + 1 × 10 + 8 = ........................................................................................

50 Oasis School Mathematics Book-5