Vedanta Excel in Mathematics Book 5 Final (2078) NEW

The Metric Measurement System

6. Let's convert into the decimal of higher units. Then add or subtract.

a) 7 cm 6 mm + 9 cm 8 mm b) 5 m 65 cm + 6 m 86 cm
c) 4 km 730 m + 8 km 690 m d) 14 cm 3 mm – 6 cm 7 mm
e) 45 m 36 cm – 27 m 52 cm f) 20 km 340 m – 8 km 650 m
7. Let's multiply or divide.

a) 3 × (4 cm 3 mm) b) 5 × (3 m 60 cm) c) 4 × (2 km 700 m)

d) (6 cm 4 mm) ÷ 4 e) (7 m 20 cm) ÷ 6 f) (25 km 480 m) ÷ 8

8. Let's convert into the decimal of higher units. Then multiply or divide.

a) 4 × (5 cm 6 mm) b) 3 × (4 m 75 cm ) c) 5 × (3 km 450 m)

d) (7 cm 2 mm) ÷ 6 e) (9 m 64 cm) ÷ 4 f) (2 km 340 m) ÷ 3

9. a) A rubber is 12 cm 8 mm long and it is stretched by 5 cm 7 mm. Find the
length of the stretched rubber.

b) A building has two storeys. The height of the irst storey is 4 m 30 cm and
the height of the second storey is 5 m 75 cm. Find the total height of the
building in decimal of metres.

c) The height of an electric pole is 13 m 50 cm. If it is 11 m 90 cm high above
the ground, ind the length of it's underground part.

d) When Bhurashi eats a certain part of a 8 cm 5 mm long chocolate bar,
4 cm 7 mm long part is remained. Find the length of the part eaten by her.

e) The road distance between D
the places is given in the igure
alongside. Answer the following
questions.

(i) How far is the place C from A? A 5 km 750 m B 6 km 875 m C

(ii) By how much is the place B nearer 3 km 500 m
to A than to C? E

(iii) If you travel 8 km 100 m from D to E via B, ind the distance between
B and D.

10. a) A story book is 1 cm 4 mm thick. Find the height made by 8 story books
placed one above another.

b) Bamboo is known as one of the fastest growing plants. It can grow 85 cm in
one day. How much does it grow in a week?

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The Metric Measurement System

c) Sound travels 343 m in one second. Calculate the distance travelled by
sound in one minute. Express your answer in decimal of kilometres.

d) The distance between two towns is 7.6 km. If a local bus makes 7 trips
between these towns everyday, ind the distance travelled by the bus in one
day.

11. a) The height of 6 books placed one above another is 10 cm 2 mm and the
thickness of each book is equal. Find the thickness of each book.

b) The length of the given wooden block is 1.5 m 1.5 m

and it is cut into 10 small blocks with equal

thickness. Find the thickness of each small block.
(Hint: convert 1.5 m into cm)

c) The height of each smaller cement block is 4.5 m
15 cm. How many blocks are needed to build a 15 cm
4.5 m tall wall?

d) A local bus travels 42 km 500 m in its 5 trips between two villages. How
much distance does it travel in 1 trip?

e) A car can travel 123.2 km with 8 l of petrol. How many kilometres does it
travel with 1 l of petrol?

It's your time - Project work!
12. a) Let's measure the length and breadth of your mathematics book.

(i) Find the perimeter of the surface of the book.
(ii) Find the difference of the length and breadth of the book.
b) Let's measure the thickness of your mathematics and science books.
(i) Which one is the thicker book and by how much?
(ii) Place one book above the other and ind the total thickness of two books.
c) Let's measure the length and breadth of your classroom by using a

measuring tape.

(i) Find the perimeter of the room.

(ii) Find the difference of the length and breadth of the room.

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13.5 Measurement of weight

The heaviness of an object is called its weight.

We use different weights and balances to
measure the weight of objects. Pan balance, dial
balance, spring balance, etc. are some of the
common balances.

Milligram (mg), gram (g), and kilogram (kg) are
some of the standard units of measurement of weights in Metric Measurement
System.

Let's review the relationship between the units of weight.

1 g = 1000 mg g × 1000 mg and mg ÷ 1000 g
1 kg = 1000 g kg × 1000 g and g ÷ 1000 kg

Now, let's study the following examples and learn about the conversion,
addition, subtraction, multiplication, and division of the units of weights.

Example 1: Convert a) 3 kg 375 g b) 1250 g into kg

Solution

a) 3 kg 375 g = 3 kg + 375 kg b) 1250 g = 1000 g + 250 g
1000

= 3 kg + 0.375 kg = 1 kg + 250 kg
1000

= 3.375 kg = 1 kg + 0.250 kg

= 1.25 kg

Example 2: A fruit seller sold 9 kg 850 g of lichees and 15 kg 350 g of
mangoes on a day.

(i) How much fruits did he sell altogether?

(ii) By how much did he sell more mangoes than lichees?

Solution 9 kg 850 g = 9.85 kg
(i) Weight of lichees = 9 kg 850 g

Weight of mangoes = + 15 kg 350 g 15 kg 350 g = + 15.35 kg

Total weight = 25 kg 200 g 25.20 kg

Therefore, he sold 25 kg 200 g or 25.2 kg of fruits altogether.

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(ii) Weight of mangoes = 15 kg 350 g 15 kg 350 g = 15.35 kg

Weight of lichees = – 9 kg 850 g 9 kg 850 g = – 9.85 kg

Total weight = 5 kg 500 g 5.50 kg

Therefore, he sold 5 kg 500 g or 5.5 kg more mangoes than lichees.

Example 3: The weight of a box ϔilled with 20 packets of potato chips is
Solution 1 kg 950 g and the weight of each packet of chips is 85 g.

(i) Find the weight of chips.

(ii) Find the weight of the empty box.

(i) Weight of 1 packet of chips = 85 g

? Weight of 20 packets of chips = 20 × 85 g = 1700 g = 1 kg 700 g

(ii) Weight of box with chips = 1 kg 950 g

Weight of chips = 1 kg 700 g

? Weight of the empty box = 1 kg 950 – 1 kg 700 g = 250 g

Therefore, the weight of the empty box is 250 g.

Example 4: The weight of a cake is 1 kg 200 g. If it is cut into 8 equal
pieces, ϔind the weight of each piece.

Solution

Here, the weight of each piece = 1 kg 200 g ÷ 8

= 1200 g ÷ 8

= 150 g

Therefore, the weight of each piece is 150 g.

EXERCISE 13.3
Section A - Classwork
1. Let's convert the lower units of weights.

a) 1 g = mg, 2 g = mg, 5 g = mg

b) 1.2 g = 1200 mg, 1.5 g = mg, 2.6 g = mg

c) 1 kg = g, 3 kg = g, 6 kg = g

d) 1.4 kg = 1400 g, 1.7 kg = g, 3.5 kg = g

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2. Let's say and write the answer as quickly as possible.

a) How many g and mg are in 1300 mg? g mg

b) How many g and mg are in 2800 mg? g mg
c) How many kg and g are in 1500 g? kg g

d) How many kg and g are in 3200 g? kg g

e) How many g and mg are in 5.6 g? m mg
f) How many kg and g are in 4.3 kg? m mg
3. Let's add and regroup into the higher units.

a) 300 mg + 700 mg = mg = g

b) 500 mg + 800 mg = mg = g mg

c) 600 g + 400 g = g= kg

d) 900 g + 6000 g = g= kg g

4. Let's convert into the lower units and subtract.

a) 1 g – 200 mg = mg – 200 mg = mg

b) 1 g – 600 mg = mg – 600 mg = mg

c) 1 kg – 300 g = g – 300 g = g

d) 1 kg – 950 g = g – 950 g = g

Section B

5. Let's convert the units of weight as indicated:

a) 1 g 250 mg (into mg) b) 3 g 425 mg (into mg)

c) 1 kg 375 g (into g) d) 2 kg 560 g (into g)

e) 2 g 115 mg (into g) f) 7 g 780 mg (into g)

g) 3 kg 420 g (into kg) h) 5 kg 900 g (into kg)

6. Let's add, subtract, multiply, or divide.

a) 8 kg 675 g + 6 kg 525 g b) 12 kg 870 + 20 kg 890 g
c) 10 kg 300 g – 4 kg 700 g d) 15 kg 240 g – 7 kg 550 g

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e) 5 × (3 kg 300 g) f) 8 × (4 kg 700 g)

g) (5 kg 480 g ) ÷ 4 h) (12 kg 600 g) ÷ 9

7. a) Father bought 4 kg 500 g of potatoes and 1 kg 750 g of tomatoes.

(i) Find the total weight of vegetables he bought.

(ii) How much more potatoes than tomatoes did he buy?

b) A fruit seller sold 15 kg 780 g of oranges and 18 kg 250 g of apples on a day.

(i) How much fruits did she sell altogether?

(ii) How much more apples than oranges did she sell?

c) A kangaroo and her joey together have a weight of 84 kg 350 g. The mother
kangaroo has weight of 75 kg 500 g.

(i) Find the weight of the joey

(ii) How much more weight does the mother kangaroo have than joey?

d) The total weight of a school bag and some books inside it is 2 kg 150 g. If the
weight of the empty bag is 360 g, ind the weight of the books.

8. a) A family consumes 1 kg 350 g of rice everyday. How much rice does the
family consume in a week?

b) An empty box is 550 g and it contains 12 packets of milk powder each of 450 g.
(i) Calculate the weight of the milk powder.
(ii) Find the weight of the box with milk powder.

c) The weight of box illed with 30 packets of noodles is 2 kg 925 g and the
weight of each packet of noodles is 90 g.

(i) Find the weight of noodles. (ii) Find the weight of the empty box.

9. a) 5 boiled eggs provide us about 66 g 500 mg of protein. How much protein
does 1 boiled egg provide?

b) The weight of a large sized bread is 1 kg 140 g. If it is cut into 6 equal slices,
ind the weight of each slice.

c) A bag contains 17 kg 600 g of sugar. If it is divided equally and put into 8
packets, how much sugar is there in each packet?

It's your time - Project work!

10. a) Let's make a group of 5 friends of your class. Measure your weights using a
dial balance or a digital balance. Find the difference of your weight with the
weights of your 4 other friends.

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b) Let's measure the weight of your school bags with books, exercise books,
box, water bottle, etc.

(i) How much weight do you carry everyday while coming to school?

(ii) Compare this weight with the weights of school bags carried by your any
other 5 friends.

c) Let's discuss with your parents about the estimated quantity of consumption
of rice by your family (i) on 1 day (ii) in 1 week (iii) in 1 month

d) Do you know a cup of white rice contains about 53.5 g of carbohydrates?
Let's estimate, how much carbohydrate you consume:

(i) on 1 day (ii) in 1 month (iii) in 1 year?

13.6 Measurement of capacity
The amount of liquid that a vessel can hold into it
is the capacity of the vessel.

We use jars and cylinders of different capacities
to measure the quantity of liquids.

Millilitre (ml) and litre (l) are some of the
standard units of measurement of capacities in Metric Measurement System.

Let's review the relationship between millilitre (ml) and litre (l).

1 l = 1000 ml l × 1000 ml and ml ÷ 1000 l

Now, let's study the following examples and learn about the conversion,
addition, subtraction, multiplication, and division of the units of capacities.

Example 1: Convert a) 4 l 750 ml b) 1500 ml into l.

Solution

a) 4 l 750 ml = 4 l + 750 l b) 1500 ml = 1500 l
1000 1000

= 4 l + 0.750 l = 15 l
10

= 4.75 l = 1.5 l

Example 2: A tea-stall owner uses 12 l 600 ml of milk and 6 l 700 ml of
water to make tea everyday. What amount of tea does
she make each day?

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Solution Another process
Amount of milk in tea = 12 l 600 ml 12 l 600 ml = 12.6 l

Amount of water in tea = + 6 l 700 ml 6 l 700 ml = + 6.7 l

Amount of tea = 19 l 300 ml 19.3 l

Therefore, she makes 19 l 300 ml or 19.3 l of tea each day.

Example 3: A glass can hold 175 ml of water. If 10 glasses of water can ϔill
a jug, ϔind the capacity of the jug.

Solution
Amount of water in 1 glass = 175 ml

Amount of water in 10 glasses = 10 × 175 ml = 1750 ml = 1 l 750 ml

Capacity of the jug = 10 glasses of water

= 1 l 750 ml = 1.75 l

Therefore, the capacity of the jug is 1 l 750 ml or 1.75 l.

Example 4: The capacity of a bucket is 7 l 200 ml and 6 jugs of water can
ϔill it completely. Find the capacity of the jug.

Solution 6 7 l 200 ml 1l 200 ml
Here, the capacity of the jug = 7 l 200 ml ÷ 6 –6
1 l 200
= 1 l 200 ml 1200 ml
= 1.2 l
– 1200 ml
0

EXERCISE 13.4
Section A - Classwork

1. Let's convert higher to lower or lower to higher units of capacity.

a) 1 l = ml, 3 l = ml, 6 l = ml

b) 1.2 l = ml, 2.5 l = ml, 4.8 l = ml

c) 2000 ml = l, 4000 ml = l, 7000 ml = l

d) 1300 ml = 1.3 l, 2400 ml = l, 3500 ml = l

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2. Let's add, subtract, or multiply as quickly as possible.

a) 400 ml + 600 ml = ml =l

b) 800 ml + 700 ml = ml = l ml

c) 1 l – 300 ml = ml – 300 ml = ml

d) 1 l – 600 ml = ml – 600 ml = ml

e) 4 × 300 ml = ml = l ml

f) 7 × 400 ml = ml = l ml

Section B

3. Let's convert the units of capacity as indicated.

a) 1 l 125 ml (into ml) b) 2 l 250 ml (into ml)
c) 1 l 475 ml (into l) d) 3 l 500 ml (into l)
e) 0.345 l f) 1.7 l
(into ml) (into ml)
g) 1300 ml (into l) h) 2800 ml (into l)

4. Let's add, subtract, multiply, or divide.

a) 7 l 550 ml + 5 l 750 ml b) 14 l 480 ml + 10 l 840 ml

c) 9 l 200 ml – 4 l 300 ml d) 20 l 500 ml – 8 l 750 ml

e) 6 × (2 l 400 ml) f) 9 × (3 l 500 ml)

g) (4 l 200 ml) ÷ 3 h) (10 l 400 ml) ÷ 8

5. a) In an average, a cow gives 7 l 450 ml of milk in the morning and 6 l 650 ml
of milk in the evening.

(i) How much milk does the cow give on a day?

(ii) How much more milk does the cow give in the morning?

b) A painter mixed 3 l 750 ml of red and 2 l 500 ml of blue paint to make purple
paint.

(i) How many litres of purple paint did he make?

(ii) How much more red paint did he used?

c) After travelling a certain distance a motorbike has 9 l 750 ml of petrol left
in its tank. When 5 l 750 ml of petrol was illed, the tank became full. Find

the capacity of the motorbike tank.

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6. a) A glass can hold 150 ml of water. If 10 glasses of water can ill a jug, ind the
capacity of the jug.

b) A doctor prescribed 20 ml of medicine three times a day for a month to a
patient. How much medicine did he take in 30 days.

c) A family consumes 3 packets of milk everyday. If a packet contains 500 ml
of milk, how much milk does the family consume in a week?

d) A girl drinks 8 glasses of water everyday. If the capacity of each glass is
275 ml, how much water does she drink in a day?

7. a) The capacity of a jug is 1 l 800 ml and 5 glasses of water can ill it completely.
Find the capacity of the glass.

b) A vessel can hold 10 l 500 ml of milk. How many jars each of 1 l 500 ml
capacity are needed to ill the vessel completely?

c) A dairy ills milk in plastic packets each of capacity 500 ml. How many
packets are required to ill 100 l of milk?

d) A sick woman bought a 180 ml bottle of medicine with it a 5 ml spoon. The
label says 'one spoonful three times a day'. How long will the bottle last?

e) A tea-stall owner uses 10 l 650 ml of milk and 4 l 750 ml of water to make
tea everyday.

(i) What amount of tea does she make each day?

(ii) If the capacity of her each serving cup is 110 ml, how may cups of tea
does she make each day?

It's your time - Project work!

8. a) Let's take a water bottle of 1 l capacity.

(i) Gently pour as many glasses of water into the bottle as to ill it completely
and estimate the capacity of the glass.

(ii) Gently pour as many cups of water into the bottle as to ill it completely
and estimate the capacity of the cup.

b) Boys and girls of ages 8 to 12 years need 2.2 l of water everyday. Estimate,
how much water do you drink

(i) on 1 day (ii) in 1 week (iii) in 1 month (iv) in 1 year?

c) Do you know 100 ml of milk provides about 130 mg of calcium? Our body

needs calcium to build and maintain strong bones. Estimate how much

milk you consume (i) on 1 day (ii) in 1 week (iii) in 1 month. Then

calculate the amount of calcium you are getting on 1 day, in 1 week and in

1 month. "

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Unit Perimeter, Area, and Volume

14

14.1 Perimeter of plane shapes - Looking back

Classwork - Exercise

1. Each room of the graphs represents 1 cm length. Let's say and write how
far Mickey Mouse has to walk to get round each igure.

cm cm cm
cm cm

cm cm

2. Let's say and write the perimeter of these plane igures.

a) b) 3 cm c) 5 cm

4 cm 6 cm 2 cm 2 cm

3 cm 3 cm

4 cm 3 cm 5 cm

Perimeter of triangle Perimeter of square Perimeter of rectangle

= ==

Thus, the perimeter of a plane igure is the distance (or length) all the way
round the igure.

Therefore, perimeter of a triangle = total of lengths of it's 3 sides

Perimeter of a quadrilateral = total lengths of it's 4 sides

In this way, to ind perimeter of a plane igure, we should ind the total length of
the boundary of the igure.

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Perimeter, Area and Volume R

14.2 Perimeter of triangle and quadrilateral

C
S

AB PQ

Perimeter of triangle ABC Perimeter of quadrilateral PQRS
= AB + BC + CA = PQ + QR + RS + SP

14.3 Perimeter of rectangle and square

The lengths of a rectangle All four sides of a square
are equal and its breadths are equal.
are also equal.

DlC HG

bb

AlB EF

Perimeter of rectangle ABCD Perimeter of square EFGH

= AB + BC + CD + DA = EF + FG + GH + HE

=l+b+l+b =l+l+l+l

= 2l + 2b = 4l

= 2(l + b)

In this way, perimeter of a rectangle = 2(l + b)

And, perimeter of a square =4×l

Example 1: On sports day in school, Anita Tamang completed 5 rounds
around a rectangular ground of length 50 m and breadth
40 m. Find the distance she covered.

Solution

Here, length of the ground (l) = 50 m 40 m

Breadth of the ground (b) = 40 m

Perimeter of the ground = 2(l + b) 50 m

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Perimeter, Area and Volume

= 2(50 + 40)

= 2 × 90

= 180 m

The distance covered by her in 1 round = 180 m

? The distance covered by her in 5 rounds = 5 × 180 m = 900 m

Example 2: If the perimeter of a square garden is 140 m, ind the length

of the garden.

Solution l

Perimeter of the square garden = 140

or, 4 × l = 140 l 4l = 140 m l
l
or, l = 140
4
or, l = 35 m.

Hence, the length of the garden is 35 m.

Example 3: The length of a rectangle is 6 cm and it's perimeter is 20 cm.
Find it's breadth.

Solution

The length of the rectangle (l) = 6 cm

The perimeter of the rectangle = 20 cm Shorter process
2(l + b) = 20
or, 2(l + b) = 20

or, 2(6 + b) = 20 or, (6 + b) = 20
2
or, 12 + 2b = 20
or, 6 + b = 10

or, 2b = 20 – 12 = 8 or, b = 10 – 6

or, b = 8 = 4 = 4 cm
2

Therefore, the breadth of the rectangle is 4 cm.

EXERCISE 14.1
Section A - Classwork
1. Let's say and write the answers as quickly as possible.

a) If the length of sides of a triangle are a cm, b cm, and c cm respectively, the
perimeter of the triangle =

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b) If the sides of a triangle are 2 cm, 3 cm, and 4 cm, perimeter =
c) If the length of sides of a quadrilateral are w cm, x cm, y cm and z cm

respectively, its perimeter =

d) If the sides of a quadrilateral are 3 cm, 4 cm, 5 cm, and 6 cm respectively, its
perimeter =

e) If l = 4 cm and b = 3 cm, perimeter of rectangle =
f) If l = 5 cm, perimeter of a square =

Section B
2. Let's ind the perimeter of the following plane shapes.

a) 4 cm b) 4 cm c) 2 cm 2 cm
3 cm
2.5 cm 2 cm

5 cm 4 cm
3 cm

1cm
1cm
3 cm

3 cm 6 cm

3. Let's identify whether these igures are rectangle or square then ind

their perimeters by using formulae.

a) 4 cm b) 3 cm c) 2 cm d) 2 cm
3 cm
3 cm
3 cm

3 cm
3.5 cm
3.5 cm

2 cm
2 cm
3 cm 2 cm
4 cm
2 cm

4. Let's ind the perimeters of these plane shapes of the given sides.

a) Triangles : (i) 5 cm, 7 cm, 9 cm (ii) 4.5 cm, 6.5 cm, 7 cm

b) Quadrilaterals: (i) 3 cm, 4 cm, 5 cm, 6 cm (ii) 4.6 cm, 5.8 cm, 7.6 cm, 10 cm

c) Rectangles: (i) l = 8.7 cm, b = 5.3 cm (ii) l = 10.4 cm, b = 7 cm
5. a) A garden is in the shape of a triangle. The length of its sides are 20 m,

25.5 cm and 35.5 m.

(i) Find the perimeter of the garden.

(ii) Find the length of wires required to fence around it with 3 rounds.

b) A piece of land is in the shape of a quadrilateral. The length of its sides are
40 m, 50 m, 36 m, and 64 m.

(i) Find the perimeter of the land.
(ii) Find the length of wires required to fence around it with 2 rounds.

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6. a) A rectangular swimming pool is 45 m long and 25 m wide. Find the following:

(i) The perimeter of the pool.

(ii) If a boy swims along its edges with 4 rounds, ind the distance covered
by the boy.

b) A rectangular garden is 60 m long and 40 m broad.
(i) Calculate the perimeter of the garden.
(ii) Find the length of fencing wires required to fence around it with 5 rounds.

c) A square ground is 40 m long. (i) Find its perimeter (ii) If a girl runs
around it and makes 3 rounds, ind the distance covered by her.

7. a) The perimeter of a square park is 200 m. Find the length of the park.

b) When Ramesh completes one round around a square ield he covered 180 m.

(i) What is the perimeter of the ield? (ii) Find the length of the ield.

c) Length of a rectangle is 8 cm and it's perimeter is 26 m. Find the breadth of
the rectangle.

d) A rectangular ground is 70 m wide and its perimeter is 300 m. Find the
length of the ground.

8. a) A rectangle is x cm wide and 2x cm long. Its perimeter is 24 cm.

(i) Make an equation and solve it to ind the value of x.

(ii) Find the length and breadth of the rectangle.

b) A rectangular pond is x m broad and 3x m long. If its perimeter is 160 m,
ind its length and breadth.

c) The length of a rectangular surface is two times its breadth and its perimeter
is 30 cm. Find its length and breadth.

d) A rectangle is three times longer than its breadth and its perimeter is
32 cm. Find its length and breadth.

It's your time - Project work!

9. a) Let's measure the length and breadth of the following objects. Then, ind the
perimeter of the surface of each object.

(i) Surface of your math book (ii) Surface of your exercise book

(iii) Surface of your desk (or chair)

(iv) Surface of white (or black) board of your class.

b) Let's draw a pair of triangles and a pair of quadrilaterals by using a ruler.
Measure the length of sides of each triangle and quadrilateral. Then, ind
the perimeter of each igure. Let's compare the perimeters of your igures
to the perimeters of your friends' igures.

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14.4 Area of plane shapes - Looking back

Classwork - Exercise

1. The area of each square room of the graphs represents 1 cm2
(1 square cm). Let's say and write the area of each plane igure.

Area = Area =

Area =

Area = Area = Area =

Thus, the area of a plane shape is the space (or region) covered by the surface of
the shape. We usually measure area in square centimetre (cm2) or in square
meter (m2).

14.5 Area of rectangle and square - Looking back

Classwork - Exercise

1. The area of each square room is A
1 cm2. Let's read the instructions and
complete the following table. C

a) Count the number of square rooms
enclosed by each rectangle and
square.

b) Write the area of each rectangle and B
square

c) Measure the length and breadth of
each rectangle and square.

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Perimeter, Area and Volume

d) Investigate the formula to ind the area of rectangle and square.

Rectangle/ No. of Area Length Breadth l×b Area of
Square rooms (l) (b) rectangle
enclosed

A

B
C

2. The area of each square room is represented by 1 cm2. Let's write the
answer in the blank spaces.

a) D C Area of the rectangle ABCD =
Al b No. of rooms along length (l) =

B No of rooms along breadth (b) =

l×b= is the area of square.

b) S R Area of the rectangle PQRS =
Pl
b No. of rooms along length (l) =

Q No. of rooms along breadth (b) =

l×b= is the area of rectangle.

Thus, area of rectangle = length × breadth = l × b

In the case of a square, its length and breadth are equal. l2 l
∴ Area of square = length × breadth l

= length × length = l2

Now, let's study the following examples and learn to solve problems related to
area of rectangles and squares.

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Perimeter, Area and Volume

Example 1: A rectangular ground is 35 m long and 28 m broad. Find its area.
Solution

Length of the ground (l) = 35 m
Breadth of ground (b) = 28 m
Area of the ground (A) = l × b = 35 m × 28 m

= 980 m2
Therefore, area of the ground is 980 m2.

Example 2: A square room is 9 m long. Find the area of its ϔloor. Also, ϔind the
area of carpet required to cover the ϔloor.

Solution

Length of the square room (l) = 9 m

Area of the loor of the room = l2

= (9m)2 = 81 m2

Therefore, area of the loor of the room is 81 m2.
Also, the area of carpet = the area of the loor = 81 m2.
Example 3: A rectangle is 6 cm long and it's area is 24 cm2. Find its breadth.

Solution 24 cm2 b
Length of the rectangle (l) = 6 cm
Area of the rectangle = 24 cm2
l × b = 24

or, 6 × b = 24

or, b = 24 = 4 cm 6 cm
6

Therefore, the breadth of the rectangle is 4 cm.

Example 4: If the perimeter of a square is 28 cm, ϔind its area.

Solution

Perimeter of the square = 28 cm

4 × l = 28 cm

or, l= 28 = 7 cm 28 cm
Now, 4 l

area of the square = l2 = (7cm)2 = 49 cm2

Therefore, area of the square is 49 cm2.

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Example 5 : In the given ϔigure, ϔind the areas of bigger and smaller rectangles.
Then, ϔind the area of the shaded region.

Solution
2cm
Length of bigger rectangle (l) = 8 cm and the breadth (b) = 6 cm 4 cm 6 cm
Area of bigger rectangle = l × b = 8 cm × 6 cm = 48 cm2 8 cm
Length of smaller rectangle (l) = 4 cm and the breadth (b) = 2 cm

Area of smaller rectangle = l × b = 4 cm × 2 cm = 8 cm2

Now, area of the shaded region

= Area of bigger rectangle - Area of smaller rectangle

= 48 cm2 - 8 cm2 = 40 cm2

Therefore, the area of the shaded region is 40 cm2

EXERCISE 14.2

Section A - Classwork

1. The area of each square room is 1 cm2 and each half of the square room is
1
2 cm2. Let's say and write the area of each plane shape.

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2. Let's say and write the area of these rectangles or squares as quickly as
possible.

a) b) c)

3 cm 3 cm 2 cm

4 cm 3 cm 6 cm

Area = Area = Area =

3. Let's say and write the area of rectangles and squares as quickly as
possible.

a) l = 4 cm, b = 2 cm, area = b) l = 5 cm, area =

c) l = 7 cm, b = 5 cm, area = d) l = 7 cm, area =

Section B
4. Let's answer the following questions.

a) De ine perimeter of a plane shape.
b) Write the formula to ind the perimeter of rectangle and square.
c) De ine area of a plane shape ? In what way perimeter is different from area?
d) Write the formula to ind the area of rectangle and square.
5. Let's ind the area of these rectangular or square surface of objects.

a) b) 6 cm 6 cm c) d)50 cm
18 cm
15 cm 70 cm

20 cm 18 cm

6. Let's ind the area of rectangles and squares.

a) l = 6 cm, b = 4.5 cm b) l = 8 cm c) l = 7.4 cm, b = 5 cm

d) l = 10 cm, b = 8.2 cm e) l = 12 m f) l = 15.6 m, b = 12 m

7. Let's ind the unknown length of side of these rectangles and squares.

a) b) c)

Area = 30 cm2 Area=16cm2 l = ? Area = 63 cm2 b=7cm
b=?

l = 6 cm l=? l=?
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8. a) A rectangular garden is 40 m long and 27 m wide. Find its area.

b) The length and breadth of a rectangular table is 2 m and 1.5 m respectively.
Find the area of its surface.

c) A rectangular room is 8 m long and 6 m broad. Find the area of its l o o r .
Also ind the area of carpet required to cover the loor.

d) A square park is 75 m long. Find the area of the park.

e) The length of a square banquet hall is 30 m and 1 of its area is occupied by
a banquet stage. Find the area of the stage. 4

9. a) A rectangle is 10 cm long and its area is 70 cm2. Find its breadth.

b) A rectangular ield is 25 m broad and its area is 1000 m2. What is the length
of the ield ?

c) If the area of a square is 81 cm2, ind its length.

d) Find the width of square loor of a room which has the area of 64 sq. m.
10. a) The perimeter of a square is 28 cm. (i) its length (ii) Find its area.

b) A square garden has perimeter of 80 m, ind its area.

c) The length of a rectangle is 9 cm and its perimeter is 30 cm.

(i) Find its breadth. (ii) Find its area.

d) The perimeter of a table tennis board is 9 m and it is 1.5 m broad.

(i) Find its length. (ii) Find its area.

11. Let's ind the area of the shaded regions. (Hint : Find the area of bigger
and smaller rectangles. Then subtract smaller area from bigger one.)

a) b) 6m 8m c) 9cm
6m
4cm 8 cm 7 cm
6 cm 8m
3 cm
10 cm 5 cm

It's your time - Project work !

12. a) Let's measure the length and breadth of the given objects using a
30 cm- ruler. Then ind the area of the surface of each object.

Objects Length (l) Breadth (b) Area
Maths book
Exercise book

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b) Let's measure the length and breadth of the surface of the following objects
using a measuring tape. Then, ind the area of the surfaces.

Objects Length (l) Breadth (b) Area
Desk / Table

White / Black Board

Classroom

14.6 Area of land
We usually measure area of land in our local Nepali system. We use the units

of Bigha (la3f) in the Terai region and Ropani (/fk] gL) in the Hilly region.

Let's study and learn about the relationship between different local units of
measurement of area of land.
Measurement of area of land in Bigha System:

1 bigha (la3f) = 20 khattha (s¶f) and 1 bigha = 6772.63m2 = 72900 Sq. ft

1 kattha (s¶f) = 20 dhur (w/' ) and 1 katta = 338.63 m2 = 3645 Sq. ft
1 dhur (w'/) = 16.93 m2 = 182.25 Sq. ft
Measurement of area of land in Ropani System:

1 ropani (/f]kgL) = 16 aana (cfgf) and 1 ropani = 508.72 m2 = 5476 Sq.ft
1 aana (cfgf) = 4 paisa (k;} f) and 1 aana = 31.80 m2 = 342.25 Sq.ft

1 paisa (k};f) = 4 daam (bfd) and 1 paisa = 7.95 m2 = 85.56 Sq. ft

1 daam (bfd) = 1.99 m2 = 21.39 Sq. ft

Do you know?
1 bigha of land = 13.31 ropani

Now, let's study the examples and learn about conversion, addition, and
substraction of units of area of land.
Example 1 : Convert a) 1 bigha 10 kattha into khattha

b) 2 ropani 6 aana into aana.
Solution

a) 1 bigha 10 kattha = 20 kattha + 10 kattha = 30 kattha

b) 2 ropani 6 aana = 2 × 16 aana + 6 aana = (32 + 6) aana = 38 aana

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Example 2 : Add a) 15 kattha + 12 kattha
b) 12 aana + 8 aana and express into higher units.

Solution
a) 15 kattha + 12 kattha = 27 kattha = 20 kattha + 7 kattha

= 1 bigha 7 kattha

b) 12 aana + 8 aana = 20 aana = 16 aana + 4 aana = 1 ropani 4 aana

Example 3 : Subtract a) 1 bigha - 6 kattha b) 1 ropani – 10 aana
Solution
a) 1 bigha - 6 kattha = 20 kattha - 6 kattha I got it!
1 bigha = 20 kattha
= 14 kattha
b) 1 ropani - 10 aana = 16 aana - 10 aana and
1 ropani = 16 anna!!
= 6 aana

Example 4 : If the cost of 1 bigha land is Rs 5,90,000 ϔind the cost of

a) 1 kattha b) 8 kattha of land.

Solution

Cost of 1 bigha land = Rs 5,90,000

Cost of 20 kattha land = Rs 5,90,000

a) Cost of 1 kattha land = Rs 5,90,000 = Rs 29,500
20

b) Cost of 8 kattha land = 8 × Rs 29,500 = Rs 2,36,000

Therefore, the cost of 1 kattha of land is Rs 29,500 and 8 kattha is Rs 2,36,000.

EXERCISE 14.3
Section A - Classwork
1. Let's say and write the answers as quickly as possible.

a) 1 bigha = kattha b) 1 kattha = dhur

c) 1 ropani = aana d) 1 aana = paisa

e) 1 bigha = m2 = Sq. feet

f) 1 kattha = m2 = Sq. feet

g) 1 ropani = m2 = Sq. feet
Sq. feet
h) 1 aana = m2 =
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Section B

2. Let's convert into as indicated.

a) 1 bigha 6 kattha (into kattha) b) 2 bigha 12 kattha (into kattha)

c) 1 kattha 8 dhur (into dhur) d) 3 kattha 10 dhur (into dhur)

e) 1 ropani 9 aana (into aana) f) 2 ropani 4 aana (into aana)

g) 25 kattha (into bigha and kattha) h) 55 kattha (into bigha and kattha)

i) 20 aana (into ropani and aana) j) 42 aana (into ropani and aana)
3. a) If the cost of 1 bigha of land is Rs 4,50,000, ind the cost of

(i) 1 kattha (ii) 6 kattha of the land

b) If the cost of 1 ropani of land is Rs 3,20,000 , ind the cost of

(i) 1 aana (ii) 8 aana of the land

c) If the cost of 1 kattha of a land is Rs 25,000, ind the cost of

(i) 1 bigha (ii) 3 bigha of land

d) If the cost of 1 aana of a land is Rs 30,000, ind the cost of

(i) 1 ropani (ii) 4 ropani of the land

It's your time - Project work !

4. a) What type of system of measurement of area of land is in your locality?
b) In how much area is your school situated ? Discuss with your teacher and
answer it.
c) In how much area is your house situated ? Discuss with your parents and
answer it.
d) Name the ministry of the Government of Nepal which is responsible in
land management.

14.7 Volume of solids - space occupied by solids (Review)

Let's take a full glass of water. Immerse a stone into the water and observe
what happens. Now, let's discuss the answers of these questions.
a) What happened when the stone is immersed into the water?
b) What caused the water to over low ?
c) Why did the water over low ?

A stone is a solid object. When it is immersed
into water, it occupies some space in the water.
The space is provided by the over lowing water.

The space occupied by a solid object is the volume of the object.

We measure volume of solids in cubic centimetre (cm3) or in cubic meter (m3).
Volume of liquid is measured in millilitre (ml) or in litre (l).

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14.8 Volume of Cube and Cuboid

The length, breadth and height of the given cube is 1 cm each.

? Its volume = 1 cu. cm (cm3) h =1cm

= 1 cm × 1 cm × 1 cm

= length × breadth × height l =1cm b=1cm
= l × l × l = l3

In this way, we ind the volume of a cube by using the formula l3.

Again, the given cuboid is made up of 6 cubes. h=1cm

Each cube has volume of 1 cm3.

? Volume of the cuboid = 6 cm3 cm =12ccmm
= 3 cm × 2 cm × 1 cm b
1 cm 1 cm 1 cm 1

l = 3 cm

= length × breadth × height = l × b × h

Thus, we ind the volume of a cuboid by using the formula l × b × h.

A cuboid is also called a rectangular solid.

Now, let's study the following examples and learn more about the volume of
solids.

Example 1: A cubical block is 5 cm long, ϔind its volume. 5 cm 5 cm
How much space does it occupy ? 5 cm

Solution
Length of the cubical block (l) = 5 cm

Volume of the block = l3

= (5cm)3 = 5 cm × 5cm × 5 cm = 125 cm3

Therefore, volume of the cubical block is 125 cm3 and it occupies a space of
125 cm3.

Example 2: A rectangular wooden block is 5 cm long, 3 cm broad and
2 cm high. Find its volume. How much water does it displace
when it is immersed into water ?

Solution

Length of the block (l) = 5 cm, breadth (b) = 3 cm and height (h) = 2cm

Volume of the block = l × b × h 2 cm

= 5 cm × 3 cm × 2 cm

= 30 cm3. 5 cm 3 cm
? Volume of the wooden block is 30 cm3.

Also, volume of water = Volume of the block = 30 cm3.

Therefore, it displaces 30 cm3 of water.

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Example 3 : If the volume of a cube is 64 cm3, ϔind its length.
Solution

Volume of the cube = 64 cm3 4 64
or, l3 = (4cm)3 4 16
or, l × l × l = 4 cm × 4 cm × 4 cm
or, length (l) = 4 cm 4
64 = 4 × 4 × 4
Therefore, length of the cube is 4 cm.

Example 4: A cuboid is 8 cm long, 5 cm broad and its volume is
160 cm3. Find it's height.

Solution

Length of the cuboid (l) = 8 cm and breadth (b) = 5 cm

Volume of the cuboid = 160 cm3 h
l × b × h = 160

or, 8 × 5 × h = 160 8 cm 5 cm
or, 40 h = 160

or, h = 160 = 16 = 4 cm
40 4

Therefore, the height of the cuboid is 4 cm.

EXERCISE 14.4
Section A - Classwork
1. Volume of each cube is 1 cm3. Let's say and write the volume of the solids.

2. Let's say and write the volume of cubes and cuboid quickly.

a) l = 2 cm b) l = 3 cm, b = 2 cm, h = 1 cm

Volume of cube = Volume of cuboid =

c) l = 4 cm, b = 3 cm, h = 1 cm d) l = 3 cm

Volume of cuboid = Volume of cube =
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3. Let's say and write the answers as quickly as possible.

a) If a piece of stone displaces 10 cm3 of water, its volume = , it

occupies a space of .

b) If a cube has volume of 8 cm3, it displaces water of volume , it

occupies a space of .

c) A brick occupies a space of 1000 cm3. Its volume = . It displaces

water of volume .

Section B
4. Let's answer the following questions.

a) De ine volume of a solid.
b) How is volume different from area ?
c) Write the formula to ind volume of a cube.
d) Write the formula to ind volume of a cuboid.
5. Let's ind the volume of the following solids.

a) b) c)

6. Let's calculate the volume of the following cubes and cuboids.

a) b) c) d)

3 cm 3 cm 4 cm 4 cm
4 cm
3 cm 3 cm 2 cm 4 cm
3 cm
15cm 6.5 cm
2.5 cm

40 cm
7. Let's ind the volume of these solid objects.

a) b) c)

15cm 15cm 10cm 80.5 cm 75 cm 12 cm
10cm
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8. Let's ind the volume of each of these cubes.

a) l = 3 cm b) l = 4 cm c) l = 5 cm d) l = 6 cm

e) l = 8 cm f) l = 10 cm g) l = 12 cm h) l = 20 cm

9. Let's ind the volume of each of these cuboids.

a) l = 5 cm, b = 3 cm, h = 2 cm b) l = 4 cm, b = 3 cm, h = 6.5 cm

c) l = 7.5 cm, b = 4 cm, h = 5 cm d) l = 10 cm, b = 4.5 cm, h = 2 cm

e) l = 8 cm, b = 6 cm, h = 3.5 cm f) l = 2 m, b = 1.5 m, h = 1 m

g) l = 3.2 m, b = 2 m, h = 5 m h) l = 2 m, b = 0.5 m, h = 2 m

10. a) A cubical block is 9 cm long, ind its volume. How much space does it
occupy ?

b) A cubical metallic block is 8 cm high, ind its volume. How much water
does it displace when it is immersed into water ?

c) A juice box is 8 cm long, 5 cm wide and 15 cm tall, ind the volume of the
box. How much juice does it hold ?

d) A dictionary is 24 cm long, 18 cm broad and 5 cm thick. Find its volume.

11. a) If the volume of a cube is 216 cm3, ind its length.

b) The volume of a cube shaped candy is 27 cm3. Find its height.

c) A cuboid is 10 cm long, 8 cm broad and its volume is 480 cm3. Find its
height.

d) A chocolate bar is 15 cm long and 5 cm wide. If its volume is 150 cm3, how
much thick is the bar ?

It's your time - Project work !
12. Let's measure the length, breadth and height (or thickness) of the following

objects. Then, ind their volumes.

a) Your maths book b) Geometry box c) Toothpaste box

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Unit Geometry - Line and Angle

15

15.1 Point, Line, and line segment - Looking back

Classwork - Exercise

1. Let's say and write the correct answers as quickly as possible.

a) Which one is the curved line in the igure ? AB
b) Which one is the line segment in the igure ? PQ

c) Which one is the straight line in the igure ? P Q
d) Name the points in the line segment PQ.
and

2. Let's name the vertical, horizontal, and slanting line segments.

a) is a vertical line segment. CB A

b) is a horizontal line segment. AB C
c) is a slanting line segment.

3. Let's measure the length of these line segments as shown. Say and write
the length quickly.

a) A B b) Q

P

AB = AB = N
c) d)

X YM

XY = MN =

15.2 Perpendicular line segment A

In the given igure, line segment AB stands on another

line segment CD and makes an angle of 900. Therefore, AB

is perpendicular to CD at B. We write AB perpendicular

to CD as AB A CD. C BD

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15.3 Parallel line segments

In the given igure, the perpendicular distance between every point of

line segments PQ and RS is equal. Therefore, line segments PQ and RS are

parallel line segments. P Q

We write PQ parallel to RS as PQ // RS. Parallel

line segments are shown as . 2 cm 2 cm 2 cm
Parallel line segments never meet each other

when they are extended in either directions. R S

15.4 Intersecting line segments A C

In the given igure, line segments AB and CD are crossing

each other at the point O. Therefore, AB and CD are the O

intersecting line segments. Here, O is called the point of D B
intersection.

15.5 Angle - Review

In the given igure, straight line segments AB and BC are B C
making a corner at B. This corner is angle ABC or angle A
CBA. We write angle ABC as ‘ABC or angle CBA as ‘CBA.

We can also write ‘ABC as ‘B. The point B, at which the
angle is made, is the vertex of ‘ABC. The line segments
AB and BC are the arms of ‘ABC.

15.6 Measurement of angles outside inside
scale scale
The given igure is a protractor. We
use protractor to measure angles. It is
also used to draw angles.
Degree is the unit of measurement

of angles and it is represented by the

symbol ( ° ). We write 60 degree as

60°, 90 degree as 90°, and so on.

Now, let's study the following illustrations and learn to measure angles by

using a protractor. D

B

AO OC

In this case, we use outside scale. In this case, we use inside scale.

? ‘AOB = 75°. ? ‘COD = 60°.

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15.7 Construction of angles
Let's learn to construct angles by using a protractor.
Construct ‘ABC = 45°.

(i) Let's draw an arm AB and place C
protractor on it as shown in
igure.

(ii) Count round the edge from 0° to B A
45° and mark C.

(iii) Remove the protractor and join
BC. Now, we constructed ‘ABC = 45°.

15.8 Types of angles by their sizes.

According to the sizes of angles, they are categorised into the following
different types.

C C
C

BA BA BA
‘ABC is an acute angle. ‘ABC is a right angle. ‘ABC is an obtuse angle.

It is less than 90°. It is 90°. It is greater than 90°.

C BA BA A
‘ABC is a straight angle. BC
It is 180°. C
‘ABC is a re lex angle. ‘ABC is a complete turn
It is greater than 180°. angle. It is 360°.

Let's remember the types of angles at a glance.

Types of angles Size Examples

1 Acute angle Between 0° and 90° 10°, 25°, 30°, 75°, 89°, …

2 Right angle Exactly 90° 90°

3 Obtuse angle Between 90° and 180° 91°, 105°, 120°, 150°, 179°, …

4 Straight angle Exactly 180° 180°

5 Re lex angle Between 180° and 360° 181°, 200°, 210°, 290°, 350°, …

6 Complete turn Exactly 360° 360°

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EXERCISE 15.1
Section A - Classwork

1. Let's say and write the pair of perpendicular and parallel line segments.

R L NE

a) A B b) c) d)

HG

C D P S QK M F

2. Let's say and write the types of angles as quickly as possible.

a) b) c)

d) e) f)

3. Let's say and write the type of angle as quickly as possible.

a) An angle which is greater than 90° but less than 180° is

b) An angle which is exactly 360° is

c) An angle which is less than 90° is

d) An angle which is exactly 180° is

e) An angle which is greater than 180° but less than 360° is

f) An angle which is exactly 90° is

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4. Let's list these angles separately in appropriate types.

105°, 80°, 360°, Acute Right Obtuse Straight Re lex Complete
300°, 150°, 90°, angle angle angle angle angle turn

60°, 180°, 270°

5. Let's say and write the size of each angle. Also, write the type of angle.
a) B b) D

O A O C

‘ AOB = angle. It is an = angle.
It is an d)
c) Z

R Q P YX

It is a = angle. It is a =
e) f) angle.
B
E

AO D O

= It is an =

It is an angle. angle.

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6. Let's measure and write the size of each angle using protractor.

a) C b) R Z

c)

B A P X
Q Y
‘ABC = =
d) E = N f)
F
F e) O

D OE
M

= ==
7. Let's compare the size of each pair of angles by using '<' or ' >' symbols.

a) b) E

C

R

B AP QD CZ Y X

‘ABC ‘PQR d) R
c)
A
FE B C F
D
a) Q PE

G

Section B

8. Answer the following questions.
a) What does CD A EF mean?
b) What does PQ // RS mean?
c) What is an angle ? De ine vertex and arms of an angle? What are the vertex
and arms of ‘ XYZ ?

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d) What are acute, obtuse, and re lex angle ? Write any three examples of
these angles.

e) What is the measurement of each of these angles ?

(i) right angle (ii) Straight angle (iii) Complete turn

f) Write a pair of angles whose sum is a right angle.

g) Write a pair of angles whose sum is a straight angle.

h) The sum of x° and 35° is a right angle. Make an equation, solve it, and ind
the value of x°.

i) The sum of y° and 110° is two right angles. Make an equation, solve it, and
ind the value of y°.

j) The sum of p° and 75° is a straight angle. Make an equation, solve it, and
ind the value of p°.

9. Let's construct the following angles by using protractor.

a) 20° b) 30° c) 40° d) 45° e) 50° f) 60°

g) 75° h) 90° i) 105° j) 120° k) 150° l) 180°

m) 210° (180° + 30°) n) 270° (180° + 90°) o) 300° (180 +120°)

10. a) Let's draw AB = 4.5 cm, and draw CD A AB at D using a protractor.

b) Let's draw PQ = 5 cm, and draw RS A PQ at S using a protractor.

c) Let's draw XY = 5.4 cm, and draw PO A XY at O using a protractor.

11. Let's measure the distance between each pair of points of the following pairs
of line segments. Make tables as shown and write the measurements in the
tables. State whether the pairs of line segments are parallel.

CE G ID YM O Q Z

AF H JB WN P RX

EF GH IJ MN OP QR

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It's your time - Project work !
12. a) Let's search, ind and write :

(i) three pairs of things which are perpendicular to each other.
(Eg. length and breadth of a book)

(ii) three pairs of things which are parallel to each other.

(iii) three pairs of things which are intersecting each other.

b) Let's make a paper-clock. Rotate the hour-hand and
minute-hand. Find out what types of angles are formed at

(i) 1 : 00 (ii) 3 : 00 (iii) 6 : 00 (iv) 8 : 00 (v) 12 : 00

c) Let's cut 5 pairs of paper strips of equal size. Stick each
pair of strips using glue and make acute, right, obtuse,
straight and re lex angles.

15.9 Types of pairs of angles by their structures and properties.

According to the structures and properties, there are 4 types of pairs of

angles.

(i) Adjacent angles (ii) Vertically opposite angles

(iii) Complementary angles (iv) Supplementary angles

(i) Adjacent angles C
B
In the given igure, ‘AOB and ‘BOC have the
A
common vertex O and a common arm OB. ‘AOB and R

‘BOC are a pair of adjacent angles. O

In this igure, ‘PQR and ‘SQR are also a pair of

adjacent angles. Here, ‘PQR + ‘SQR = a straight

angle = 180° SQ P

In this case, ‘ PQR and ‘ SQR are linear pair. Therefore, if the sum of a pair
of adjacent angles is 180°, the angles are called linear pair.

(ii) Vertically opposite angles

In the given igure, straight line segments AB and CD are D B

intersecting at a point O. ‘AOC and ‘BOD are on opposite

sides of the vertex O. The pair of angles ‘AOC and ‘BOD O
are vertically opposite angles. ‘AOD and ‘COB is

another pair of vertically opposite angles. AC

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Vertically opposite angles have a common vertex but no arm is common.
Each pair of vertically opposite angles are always equal.

? ‘ AOC = ‘ BOD and ‘ AOD = ‘COB

(iii) Complementary angles R Q

If the sum of a pair of angles is a right angle ( 90° ), the

angles are called complementary angles. In the given

igure, ‘POQ + ‘QOR = 90°. Therefore, ‘POQ and ‘QOR

are complementary angles. O P

Here, ‘POQ is the complement of ‘QOR and ‘QOR is the complement of
‘POQ.

Similarly, the complement of 60° = 90° – 60° = 30° and the complement of

30° = 90° – 30° = 60°. B

(iv) Supplementary angles

If the sum of a pair of angles is a straight angle (180°), OA
the angles are called supplementary angles. In the C
given igure, ‘AOB + ‘BOC = 180°

Therefore, ‘AOB and ‘BOC are supplementary angles.

Here, ‘AOB is the supplement of ‘BOC and ‘BOC is the supplement of
‘AOB.

Similarly, the supplement of 105° = 180° – 105° = 75° and the supplement of
75° = 180° – 75° = 105°.

EXERCISE 15.2

Section A - Classwork
1. Let's ill in the blanks with the correct answers.

a) A pair of angles have a common vertex and a common arm.

b) A pair of angles have a common vertex but they don't have
common arm.

c) The sum of a pair of complementary angles is

d) The sum of a pair of supplementary angles is

e) If the sum of a pair of adjacent angles is 180°, they are called

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2. Let's say and write the answers as quickly as possible.

a) If x° and y° are linear pair of angles, then x° + y° =

b) If a° + b° = 90°, the pair of angles a° and b° are =

c) If p° and 100° are supplementary angles, p° =

d) If x° and 60° are vertically opposite angles, x° =

3. Let's identify and write the names of the given pairs of angles.

a) D b) Y

C Z

B A O are linear pair
X
and are adjacent angles
and

S Q

c) and are vertically opposite angles.

O and are vertically opposite angles.

PR

Section B

4. Let's answer the following questions.

a) Name the type of pair of angles which have a common vertex and a

common arm.

b) Name the type of pair of angles which have a common vertex but they do

not have common arm.

c) What are adjacent angles ? write with an example.

d) What are vertically opposite angles ? Write with an example.

e) What are linear pairs ? Write with an example.

f) What are complementary and supplementary angles ? Write with

examples.

5. Let's copy the given igure and table. Then, write the names of different

types of pairs of angles. AD

Pairs of Adjacent angles Pairs of Vertically opposite angles

B

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6. a) Let's ind the complements of (i) 40° (ii) 55° (iii) 30° (iv) 75°

b) Let's ind the supplements of (i) 50° (ii) 90° (iii) 110° (iv) 150°

c) Let's ind the angles which make linear pairs with the following angles.

(i) 60° (ii) 80° (iii) 105° (iv) 135°

7. Let's make equations and solve them to ind unknown angles.

a) x° and 50° are a pair of complementary angles. Find x°.

b) y° and 30° are a pair of supplementary angles. Find y°.

c) z° and 140° are a linear pair of angles. Find z°.
Let's ind the size of unknown angles marked with letters.

8. a) b) c)

x° 45° x° x° 65°

40° 25°

d) e) y° f)

60° 40° 50° y°
x° 20°

9. a) b) c) y°
120°
p° 70° 75° x°

d) x° e) f)
y° 40°
50° 40° 30° 110° x°
20°

g) h) i)

x° x° 2x° x°
y° 45° 50° y°

10. a) 80° b) p° q° c)

x° z° 105° r° a° 45°
b° c°
y°

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y° x°
d) 70° a°
b° 150°
x°
60° y°

It's your time - Project work !
11. a) Let's write the following pairs of angles and follow the instructions.

(i) Any 3 pairs of linear pair angles and show that each pair makes linear
pair.

(ii) Any 3 pairs of complementary angles and show that each pair is
complementary angles.

(iii) Any 3 pairs of supplementary angles and show that each pair is
supplementary angles.

b) Let's draw a pair of intersecting line segments. Measure each pair of
vertically opposite angles and show that each pair of vertically opposite
angles are equal.

15.10 Transversal

Let's study the following illustrations and learn about a transversal line
segment.

P B QS A
W OX
E YN
A X

CF M YP Z
D PR

Q MN is a transversal

PQ is a transversal B

AB is a transversal

In this way, a transversal is a P A P B
line segment that intersects A E E D
two line segments at two B F
DC Q
distinct points. In the given F Q

C

igure, a transversal PQ

intersects AB and CD at the points E and F.

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15.11 Pairs of angles made by a transversal

In the given igure, transversal EF intersects line segments AB and CD at the

points G and H. The transversal makes the following E

pairs of angles with the line segments AB and CD. GB

(i) Interior and co-interior angles A 43

In the igure, ‘3, ‘4, ‘5 and ‘6 are interior angles. C 56
They are made inside the line segments AB and CD. H
Here, ‘3 and ‘6 are a pair of co-interior angles. ‘4 D
and ‘5 are also co-interior angles. F
Co-interior angles means consecutive interior angles.

If the line segments AB and CD are parallel to E B
each other, the sum of each pair of co-interior G D
angles is 180°. In the igure, AB // CD, A ab
cd
? ‘a + ‘c = 180° and ‘b + ‘d = 180° CH

F

(ii) Alternate angles E
GB
In the igure, ‘3 and ‘5 are a pair of alternate angles. 43
‘4 and ‘6 are also a pair of alternate angles. A pair of
alternate angles lie on the opposite side of transversal A 56
without a common vertex. C H

F D

If the line segments AB and CD are parallel G B
to each other, each pair of alternate angles A ab D
are equal. In the igure, AB // CD,
cd
? ‘a = ‘d and ‘b = ‘c. CH

(iii) Corresponding angles A 12 B
C 43
In the igure, ‘1 and ‘5, ‘2, and ‘6 are two pairs of 56
corresponding angles. ‘3 and ‘7, ‘4, and ‘8 are also 87 D
other two pairs of corresponding angles.

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If the line segments AB and CD are parallel to ab B
each other, each pair of alternate angles are A dc D
equal.
ef
In the igure, AB // CD, C hg

∴ ‘a = ‘e, ‘b = ‘f, ‘c = ‘g and ‘d = ‘h

Pairs of angles at a glance!

Pairs of angles Diagrams and examples

Co-interior a a ab ab
angles b
b ab ab

‘a + ‘b = 180°

Alternate a a a b
Angles b b b a

‘a = ‘b

Corresponding a a a
angles b
a
b b ab ab

b

‘a = ‘b

EXERCISE 15.3

Section A - Classwork

1. Let's study the given igure. Tell and write the correct answers in the
blank spaces.

a) ‘4 and are co-interior angles.

b) ‘3 and are alternative angles. 12
c) ‘1 and are corresponding angles. 43

d) ‘3 and ‘6 are angles. 56
e) ‘4 and ‘6 are angles. 87

f) ‘2 and ‘6 are angles.

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2. Let's say and write the correct answers as quickly as possible.

a) If x° and y° are co-interior angles between two parallel lines,

x° + y° =

b) If a° and 55° are alternate angles between two parallel lines,

a° =

c) If p° and 70° are corresponding angles between two parallel lines,

p° =

Section B

3. Answer the following questions.

a) If ‘a and ‘b are a pair of co-interior angles between parallel lines, write
the relation between ‘a and ‘b.

b) If ‘x and ‘y are a pair of alternate angles between parallel lines, write the
relation between ‘x and ‘y.

c) If ‘p and ‘q are a pair of corresponding angles between parallel lines,
write the relation between ‘p and ‘q.

d) Is the sum of co-interior angles between non-parallel lines 180° ?

e) Are the alternate angles between non-parallel lines equal ? Are
corresponding angles equal ?

4. Let's copy the igures. Name the pairs of co-interior, alternate and
corresponding angles.

BA P Q N M D C

OP

CD RSQ A BE

5. a) If x° and 110° are a pair of co-interior angles between parallel lines, ind

x°.

b) If a° and 2a° are a pair of co-interior angles between parallel lines, ind a°
and 2a°.

c) If 2y° and 80° are a pair of alternate angles between parallel lines, ind the
value of y.

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d) If 3x° and 60° are a pair of corresponding angles between parallel lines,
ind the value of x.

e) For what value of a are (a + 10)° and 75° are a pair of alternate angles
between parallel lines ?

Let's ind the size of unknown angles marked with letters.

6. a) b) x° c) y° z°

70° y° x° 45°
x° Hint: x° = 45°, z = 45° and
Hint: x° + 70° = 180° 60°
Hint: y° = 60° and x° + y ° = 180°
d) x° + 60° or x° + y° = 180°
f) y°
40° e)
x° 60°
75°

a°

g) h) 35° i) 115° x°

x° y° p°
120°

j) a° k) l)

110° 125°

7. a) x° p° 80° q°
y°
65°
a° b) c)

b° y° c° b°

d) x° 40° a° d°
60°
130°
w° e) f)

x° b° y°
z° y° a°

x° z°
85°
c°

d° 70° w°

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8. a) b) z° c) d° 110° e° b°
a°
b° c° 75° y° c°

a° 60° x°

9. Let's use the relationships between different pairs of angles and prove
the following relations.

a) z y b) z c) y

b

x c ax z

Prove that ‘x + ‘z = 180° Prove that ‘b = ‘c Prove that ‘y = ‘z

Hint: ‘x = ‘y
‘y + ‘z = 180°

It's your time - Project work !

10. a) Let's write bigger size of capital letters of English alphabets from A
to Z on a chart paper. Then, underline those letters in which you ind
co-interior, alternate, and corresponding angles. Also, mark these angles
in the letters. For example :

corresponding angles co-interior angles

b) Let's cut a few number of paper strips. Stick the strips to form

shapes using glue. Then, mark co-interior, alternate, and corresponding

angles separately. "

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Unit Geometry - Plane Shapes

16

16.1 Plane shapes - Looking back

Classwork - Exercise

1. Let's say and write the answers as quickly as possible.
a) What is the shape of a clothes hanger ?

b) What is the shape of set-squares ?

c) What is the shape of the surface of maths book ?

d) What is the shape of the surface of a die ?

e) What is the shape of the surface of a coin ?

Thus, triangle, rectangle, square, circle are plane shapes. They are also
called plane igures. A triangle has triangular surface; a rectangle has
rectangular; a square has square; and a circle has circular surfaces.

A plane igure is a 2-dimensional (2-D) shape because it has two
dimensions: length and breadth.

16.2 Triangle A
C
We write the given triangle ABC as ∆ABC. We use the symbol B
'∆' for the world 'triangle'. AB, BC, and CA are 3 sides of
∆ABC. ‘A, ‘B, and ‘C are 3 angles of ∆ABC. A, B, and C are
3 vertices of ∆ABC.

16.3 Types of triangle by sides
Let's take three sets of pencils (or toothpicks or straws, etc.) of the following

lengths:

set 1 set 2 set 3
All of equal length Two of equal length None of equal length

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Now, let's arrange each set of these pencils ot make three different shapes of
triangles.

Triangle formed from set 1 Triangle formed from set 2 Triangle formed from set 3
Triangle with all sides equal Triangle with two sides equal Triangle with none of the
sides equal
p p
p
Equilateral triangle Isosceles triangle
Scalene triangle

In this way, there are three types of triangles according to the length of sides
of triangles.

(i) Equilateral triangle (ii) Isosceles triangle and (iii) Scalene triangle

(i) Equilateral triangle C

All three sides of an equilateral triangle are equal. 60°
Each angle of an equilateral triangle is 60°. In the
given equilateral triangle ABC, 60° 60°
AB = BC = CA and ‘A = ‘B = ‘C = 60°. A B

(ii) Isosceles triangle

Any two sides of an isosceles triangle are equal. In the given R

isosceles triangle PQR, PR = QR. Here, the remaining side
(unequal side) PQ is called the base of the isosceles triangle
PQR. ‘PQR and ‘QPR are called the base angles of ∆PQR.

The base angles of an isosceles triangle are always equal. P Q

? ‘PQR = ‘QPR

(iii) Scalene triangle F

None of the sides of a scalene triangle are equal. In the D E
given scalene triangle DEF, the sides DE, EF and FD are
not equal to each other. Similarly, all three angles of a
scalene triangle are not equal to each other. Therefore,
‘D , ‘E, and ‘F are not equal to each other.

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16.4 Types of triangles by angles
According to the size of angles, there are three types of triangles.

(i) Acute angled (ii) Obtuse angled (iii) Right angled
triangle triangle triangle

A P X

BC QR YZ

All three angles are One angle (‘Q) is One angle (‘Y) is
acute. 'ABC is an obtuse. 'PQR is an right angle. 'XYZ is a
acute angled triangle. obtuse angled triangle. right angled triangle.

The given triangles are right angled triangles. In a right 90°– xx
angled triangle, one angle is right angle (90°) and two 90°– x x
remaining angles are acute angles.
If one of the acute angles is x°, the other will be 90°–x.

16.5 Sum of the angles of a triangle
Let's measure the size of each angle of these triangles. Then study the
following table and investigate the fact about the sum of angles of triangles.

C CC

A Fig (i) B A Fig (ii) B A Fig (iii) B

Fig. Angles Sum of 3 angles
‘C ‘A + ‘B + ‘C
‘A ‘B 30° 90° + 60° + 30° = 180°
30° 40° + 110° + 30° = 180°
(i) 90° 60° 80° 45° + 55° + 80° = 180°

(ii) 40° 110°

(iii) 45° 55°

In this way, the sum of all three angles of a triangle is always 180°.

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EXERCISE 16.1
Section A - Classwork

Let's say and write the correct answers in the blank spaces.

1. a) If two sides of a triangle are equal, it is triangle.

b) If none of the sides of a triangle are equal, it is triangle.

c) If all three sides of a triangle are equal, it is triangle.

d) Each angle of an equilateral triangle is .

e) The unequal side of an isosceles triangle is called

f) If one of the base angles of an isosceles triangle is 40°, another base angle
is

2. a) If one of the angles of a triangle is 90°, it is triangle.

b) If all three angles of a triangle are acute angles, it is triangle.

c) If one of the angles of a triangle is 105°, it is triangle.

d) If one of the acute angles of a right angled triangle is 40°, another acute
angle is

e) The sum of three angles of a triangle is

3. Let's read the given length of sides and size of angles of the following

triangles. Then, write the type of triangle below each triangle:

3.5 cm

a) b) c) 5.7 cm
3.5 cm
3.8 cm
4 cm
3.5 cm 4.2 cm

4.2 cm

4.5 cm e) f)

d) 80°
45°
120°
55°

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4. Let's write the sum of the angles of these triangles.

a) b) c) q° r°

z° c°
x° y°
p°

a° b°

x° + y° + z° = a° + b° + x° = p° + q° + r =

Section B

5. Let's answer the following questions.
a) What are the types of triangles according to the length of sides of triangles?
b) What are the types of triangles according to the size of angles of triangles?
c) What are the following triangles ? De ine them.
(i) Isosceles triangle (ii) Equilateral triangle (iii) Right-angled triangle
d) Can any triangle have two right angles ?
e) Can any triangle have two obtuse angles ?
f) If an acute angle of a right-angled triangle is a°, what is another acute
angle?
g) If the sum of any two angles of a triangle is 110°, what is the size of the
remaining angle?
Let's ind the unknown angles marked with letters.

6. a) b) c) y°

x° 60° 55°
a°
40°
(Hint: x° = 90° – 40°) b) 40° a° c)

7. a) 70°

50° x° p°
x°
8. a) b) 85° c) 30° 65°

105° 40° 55° a° y°
(Hint: x° + 105° + 40° = 180°
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