Vedanta Excel in Mathematics Book 6 Final (2078)

Algebra

d) If m = n = 2a and a = 3, find the value of mn.
e) If p = 2x, q = 3x and x = 3, find the value of 6pq .

10. Simplify. b) x(x – 1) + 3(x – 1)
a) x(x + 1) + 2(x + 1) d) a(2a + 1) – a(2a + 1)
c) y(y + 2) + 1(y + 2) f) a(a – b) – b(a – b)
e) x(x + y) + y(x + y)

9.10 Multiplication of polynomials

In the case of multiplication of two binomial expressions, each term of a binomial
is separately multiplied by each term of another binomial. Then the product is
simplified. The multiplication of any two polynomials is also worked out in the
similar process.

Worked-out examples

Example 1: Multiply a) (a + b) by (a + b) b) (2m + 3n) by (3m – 2n)
Solution:
a) By horizontal arrangement By vertical arrangement
(a + b) (a + b) = a (a + b) + b (a + b) a+b

= a2 + ab + ab + b2 ×a+b
= a2 + 2ab + b2
a2 + ab Add
+ ab + b2

a2 + 2ab + b2

b) By horizontal arrangement By vertical arrangement
(3m – 2n) (2m + 3n)= 3m (2m + 3n) – 2n(2m + 3n) 2m + 3n

= 6m2 + 9mn – 4mn – 6n2 × 3m – 2n

= 6m2 + 5mn – 6n2 6m2 + 9mn Add
– 4mn – 6n2

Example 2: Multiply (2a – 3b + 5) by (4a – b). 6m2 + 5mn – 6n2

Solution:

By horizontal arrangement By vertical arrangement
(4a – b) (2a – 3b + 5) 2a – 3b + 5
= 4a (2a – 3b + 5) – b (2a – 3b + 5)
= 8a2 – 12ab + 20a – 2ab + 3b2 – 5b × 4a – b
= 8a2 – 14ab + 3b2 + 20a – 5b 8a2 – 12ab + 20a
– 2ab + 3b2 – 5b
8a2 – 14ab + 3b2 + 20a – 5b

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Algebra

EXERCISE 9.6

General Section – Classwork

Let's study the following tricks of multiplication of two binomials.

x×x 2+3 2×3

(x + 2) (x + 3) x2 ... (x + 2) (x + 3) x2 + 5x ... (x + 2) (x + 3) x2 + 5x+ 6

y×y 4–1 4 × (– 1) y2 + 3y – 4

(y + 4) (y – 1) y2 ... (y + 4) (y – 1) y2 + 3y ... (y + 4) (y – 1)

a×a –3–5 (– 3) × (– 5)

(a – 3) (a – 5) a2 ... (a – 3) (a – 5) a2 – 8a ... (a – 3) (a – 5) a2 – 8a + 15

1. Let’s investigate the facts of tricky calculation. Tell and write the products as
quickly as possible.

a) (x + 1) (x + 1) = ............................. b) (x + 1) (x + 2) = .............................

c) (x + 2) (x + 3) = ............................. d) (x + 2) (x – 1) = .............................

e) (a + 3) (a – 2) = ............................. f) (a – 2) (a + 5) = .............................

g) (y – 3) (y – 4) = ............................. h) (y – 4) (y – 2) = .............................

Creative Section - A

2. Simplify. b) a (a + 2) – 1 (a + 2)
a) x(x + 1) + 2 (x + 1) d) a (a + b) + b (a + b)
c) x (x + 2) – 3 (x + 2) f) 2p (2p – 3q) + 3q (2p – 3q)
e) x (x + y) – y(x + y) h) 4a (3a – 5b) –3b (3a – 5b)
g) 3x (x + 2y) – y (x + 2y)

i) p (7p – 2q) + 4q (7p – 2q)

3. Multiply. b) (x + 3) (x + 4) c) (a + 2) (a – 1)
a) (x + 1) (x + 2) e) (b – 5) (b + 5) f) (p + 4) (p – 4)
d) (a – 3) (a – 2) h) (a + b) (a – b) i) (a – b) (a – b)
g) (a + b) (a + b) k) (x – y) (2x + 3y) l) (x – y) (2x – 3y)
j) (x + y) (2x + 3y) n) (5a – 2b) (5a + 2b) o) (x – 3y) (2x + 5y)
m) (3x + 4y) (3x – 4y)

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Algebra

Creative Section - B

4. Multiply. b) (x – 2) (x2 – 2x + 3) c) (x – 3) (2x2 + 3x – 4)
a) (x + 1) (x2 + x + 1) e) (3a + 2b) (2a – 4b – 5) f) (4a – 5b) (3a + 2b – 7)
d) (2a + b) (a + b + 2)

5. a) If x = (a + 2) and y = (a – 2), show that xy = a2 – 4.

b) If x = (p – 3) and y = (p + 3), show that xy + 9 = p2.

c) If a = (2x – 3) and b = (2x + 3), show that ab + 9 = x2.
4

d) If a = (p + q), b = (p – q) and c = q2 – p2 , show that ab + c = 0.

e) If x = (2a – 3b), y = (2a + 3b) and z = 9b2 – 4a2, show that xy + z = 0.

It's your time - Project work!

6. a) Let's write two expressions of the forms ax and bx + c, where a, b, and c
are integers. Find the product of your expressions.

b) Let's write any two binomial expressions of the forms (ax + b) and (cx + d),
where a, b, c, and d are integers. Find the product of your expressions.

9.11 Division of algebraic terms

Let’s study the following illustrations and investigate the rule of division of
algebraic terms.

Example 1: Divide a) 10x2 by 2x b) 15a3b4 by 5ab2

Solution: 10x2
2x
a) 10x2 ÷ 2x = 10x2 ÷ 2x = 10x2 In division of the same
2x base we should subtract
150 × x × x lower exponent from
= 2×x = 10 x2 – 1 higher exponent of the
2 same base.
= 5x
= 5x

b) 15a3b4 ÷ 5ab2 = 15a3b4
5ab2

= 135 × a ×a × a × b ×b × b × b 15a3b4 ÷ 5ab2= 15a3b4
5× a×b×b 5ab2

=3×a×a×b×b = 15a3–1 b4 – 2
5
= 3a2b2
= 3a2b2

Thus, the rule of dividing an algebraic term by another term is :

(i) Divide the coefficient of dividend by the coefficient of divisor.

(ii) Subtract the exponent of the base of divisor from the exponent of the same
base of dividend.

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Algebra

9.12 Division of polynomials by monomials

In this case, each term of a polynomial is separately divided by the monomial.
Let's study the following examples.

Example 2: Divide (a) 10x2 – 15x by 5x (b) 14a3b2 – 8a2b3 by – 2ab
Solution:

a) (10x2 – 15x) ÷ 5x

= 120x2 – 135x
5x 5x
11
= 2x2 – 1 – 3

= 2x –

9.13 Division of polynomials by binomials

While dividing a polynomial by a binomial, at first we should arrange the terms
of divisor and dividend in descending (or ascending) order of exponents of
common bases. Then, we should start the division dividing the term of dividend
with the highest exponent by the term of divisor with the highest exponent.

Let’s learn the process from the following example.

Example 3: Divide (x2 + 5x + 6) by (x + 3)

Solution: Divide x2 by x x2 ÷ x = x (In quotient)

Multiply the divisor (x + 3) by the quotient x

x(x + 3) = x2 + 3x

Subtract the product from the dividend.

x2 + 5x + 6

± x2 ± 3x

2x + 6

Again, divide the first term of the remainder by the
first term of the divisor. Continue the process till the
remainder is not divisible by divisor.

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Algebra

EXERCISE 9.7
General Section – Classwork

Let’s subtract the exponent of base of divisor from the exponent of the same
base of dividend. Tell and write the quotient as quickly as possible.

1. a) x2 ÷ x = ........ b) x3 ÷ x = ........ c) x4 ÷ x2 = ........

d) x4 ÷ x3 = ........ e) a3 ÷ a2 = ........ f) a2 ÷ a2 = ........

g) y4 ÷ y = ........ h) y4 ÷ y4 = ........ i) y5 ÷ y3 = ........

2. a) y2z2 ÷ yz = ........ b) y3z2 ÷ yz = ........ c) y2z3 ÷ yz = ........

d) a3b3 ÷ a2b = ........ e) a3b3 ÷ ab2 = ........ f) a3b3 ÷ a2b2 = ........

g) 6x3 ÷ 2x = ........ h) 6a4 ÷ 3a = ........ i) 12p6 ÷ 4p3 = ........

Creative Section - A

3. Expand the terms of dividend and divisor, then divide.

Eg. 14x4 ÷ 7x2 = 14x4 = 124 × x ×x × x × x = 2x2
7x2 7×x×x

a) x2 ÷ x b) y3 ÷y c) a4 ÷ a2 d) 6x3 ÷ 2x2

e) 6y4 ÷ 3y3 f) 16x5 ÷ 4x2 g) 24x3y4 ÷ 8x2y2 h) 30a4b4 ÷ 5a2b3

4. Divide by using rule.

Eg. 24x5 ÷ 6x3 = 24x5 = 4x5 – 3 = 4x2
6x3
a) x3 ÷ x2 b) a4 ÷ a c) 9p2 ÷ 3p
f) 14x6 ÷ (– 7x5)
d) 15m5 ÷ 5m2 e) 9y6 ÷ (– 3y4)

g) a2b2 ÷ ab h) x3y3 ÷ xy i) 3x5y4 ÷ x2y

j) 10x3y2 ÷ (– 2xy) k) – 12a4b4 ÷ 3a2b2 l) 16x8y7 ÷ (–4x6y5)
5. Divide.

a) (10x2 – 15x) ÷ 5 b) (8x2 – 16) ÷ 8
c) (a2 + a) ÷ a d) (x2 – 3x) ÷ x
e) (4b2 + 6b) ÷ 2b f) (9c3 – 6c2) ÷ 3c2

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g) (8m3 – 6m2) ÷ 2m2 h) (6n4 – 9n3) ÷ 3n2
i) (p3q – 2p2q3) ÷ pq j) (3a4b – ab4) ÷ ab

k) (3x3y2 + 6x2y3) ÷ 3xy l) (15p4q3 – 20p2q4) ÷ 5pq

m) (12a6x5 + 18a5x6) ÷ 6a3x3 n) (20b4c6 – 50b4c4) ÷ (–10b4c4)

o) (25x5y4z3 + 40x4y5z4) ÷ 5x3y4z2 p) (21m3n4p5 – 49m6n5p4) ÷ 7m2n3p4

Creative Section - B

6. Find the quotient. b) (a2 + 5a + 6) ÷ (a + 3)
a) (x2 + 3x + 2) ÷ (x + 2) d) (p2 + 9p + 20) ÷ (p + 5)
c) (m2 + 7m + 12) ÷ (m + 4) f) (b2 + 2b – 15) ÷ (b – 3)
e) (a2 + a – 6) ÷ (a – 2) h) (y2 – 4y – 21) ÷ (y – 7)
g) (x2 – 3x – 10) ÷ (x – 5)

7. a) The product of two algebraic terms is 6a3b2. If one of the terms is 3ab,
find the other term.

b) The product of two algebraic expressions is 14x3y2 – 35x2y3. If one of the
expressions is 7x2y2, find the other expression.

c) The area of a rectangle is x2 + 4x + 3 sq. units and its length is (x + 1)
units. Find its breadth.

d) The area of a rectangle is x2 + 3x – 10 square units. If its breadth is (x – 2)
units, find its length.

8. a) If a = 3x2y, b = 4xy2 and c = 2xy, find the value of ab .
c
pq
b) If p = 5a2b2, q = 4a3b3 and r = 10a4b4, show that r = 2ab.

c) If x = 12p3q4, y = 6p2q3 and z = 2p2q2, show that x + y = 2pq + 3q.
y z

It's your time - Project work!
9. Let's write the appropriate terms in the blanks to match the given quotients.

a) ......... ÷ ......... = x b) ......... ÷ ......... = x2 c) ......... ÷ ......... = x3

d) ......... ÷ ......... = 2a e) ......... ÷ ......... = 2y2 f) ......... ÷ ......... = 3p3

10. Let's write any appropriate algebraic terms in the blanks and find the
quotients.
a) ....... ÷ ....... = ....... b) ....... ÷ ....... = ....... c) ....... ÷ ....... = .......

d) ....... ÷ ....... = ....... e) ....... ÷ ....... = ....... f) ....... ÷ ....... = .......

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Unit Equation, Inequality and Graph

10

10.1 Equation - Looking back
Classwork - Exercise

1. Let's tell and write the values of the letters as quickly as possible.

a) a + 3 = 7, a = ........... b) y + 4 = 9, y = ...........

c) x + 2 = 11 , x = ........... d) y – 1 = 3, y = ...........

e) b – 5 = 8, b = ........... f) x – 2 = 7, x = ...........

g) 2x = 6, x = ........... h) 3y = 9, y = ...........

i) 7a = 21, a = ........... j) x = 2, x = ...........
m = ........... 3 z = ...........

k) m = 5, l) z = 3,
4 7

10.2 Mathematical statements
Let’s study the following illustrations and learn about mathematical statements.

3 + 4 = 7 → The sum of 3 and 4 is 7 is a mathematical statement.

10 – 4 = 6 → The difference between 10 and 4 is 6 is a mathematical statement.

5 × 2 =10 → The product of 5 and 2 is 10. is a mathematical statement.

18 = 6 → The quotient of 18 divided by 3 is 6 is a mathematical statement.
3

Thus, the statements containing the mathematical operations like addition,

subtraction, multiplication and division are called Mathematical Statements.

A mathematical statement may be true or false. For example:
The sum of 2 and 3 is 5 is a true statement.
However, the sum of 2 and 3 is 4 is a false statement.
Similarly, the product of 2 and 5 is 10 is a true statement.
But, the product of 2 and 5 is 8 is a false statement.
Such mathematical statements are called closed Mathematical statements
(or sentences).

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Equation, Inequality and Graph

10.3 Open mathematical statements
Let’s consider a mathematical statement the sum of x and 4 is 9.

It is written in mathematical sentence as x + 4 = 9

Here, until x is replaced by any number, we cannot say whether x + 4 = 9 is a
true or false statement. For example:

If x is replaced by 3, 3 + 4 = 9, which is false.

If x is replaced by 4, 4 + 4 = 8, which is false.

If x is replaced by 5, 5 + 4 = 9, which is true.

Such mathematical statements which cannot be predicated as true or false
statements (until the variable is replaced by any number) are known as Open
Mathematical Statements.

10.4 Equation and solution of an equation
Let’s consider an open mathematical statement x + 4 = 9.

This open statement can be true only for a fixed value of x. Such open statement
containing ‘equal to’ (=) sign and can be true only for a fixed value of variable
is called an equation.

Let’s consider another equation, y – 7 = 3.

Here, y – 7 is the expression on left hand side (L.H.S.) and 3 is the expression on
right hand side (R.H.S.) of the ‘equal to’ sign.

Let’s consider an equation, x + 3 = 8.

This equation can be true only for a fixed value of x, which is 5. So, 5 is called
the solution (or root) of the equation. The process of obtaining a solution of an
equation is called solving of equation.

Let’s take a set N = {1, 2, 3, 4, 5} and an equation x + 3 = 8

Now, replace x by each element of the set N. The truth value
When, x = 1, 1 + 3 = 8 It is not true. of a variable of an
When, x = 2, 2 + 3 = 8 It is not true. equation is the
When, x = 3, 3 + 3 = 8 It is not true. solution of the
When, x = 4, 4 + 3 = 8 It is not true. equation!
When, x = 5, 5 + 3 = 8 It is true.
? x = 5 is the solution of the equation.

10.5 The facts for solving equations

Solving an equation means to find the value (solution) of the variable that makes
the equation true. We use the following facts while solving equations.

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Equation, Inequality and Graph

Fact 1 When an equal number is added to two equal quantities the sum will
Fact 2 be equal. For example,
Fact 3 If x = 2, then x + 5 = 2 + 5
Fact 4
When an equal number is subtracted from two equal quantities the
difference will be equal. For example,
If x = 8, then x – 3 = 8 – 3

When two equal quantities are multiplied by an equal number, the
product will be equal. for example,
If x = 3, then 4 u x = 4 u 3

When two equal quantities are divided by an equal number, the quotient
will be equal. For example,

If x = 24, then x = 24 .
8 8

Worked-out examples

Example 1: A = { 1, 2, 3, 4, 5} is the solution set of the variable x in the
equation x + 3 = 7. Replace the variable by each element of the
set A and state the solution of the equation.

Solution :
Here, the given solution set A ={1, 2, 3, 4, 5} and the equation is x + 3 = 7.

When x = 1, then 1 + 3 = 7 It is false
When x = 2, then 2 + 3 = 7 It is false
When x = 3, then 3 + 3 = 7 It is false
When x = 4, then 4 + 3 = 7 It is true.

∴ The required solution of the equation is x = 4.

Example 2 : Add 6 to both sides of x – 6 = 3 and find the value of x.
Solution :

x–6 =3
or, x – 6 + 6 = 3 + 6
or, x = 9
Hence, the required value of x is 9.

EXERCISE 10.1
General Section – Classwork

1. Tell and write whether the following statements are ‘true’, ‘false’ or open
statements in the blanks.

a) The sum of 3 and 5 is 8. It is ..................... statement.

b) The product of 8 and 3 is 25. It is ..................... statement.

c) The difference between x and 2 is 7. It is ..................... statement.

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Equation, Inequality and Graph

d) 3 times y is 18, It is ..................... statement. I got it!
The sum of x and
e) If x = 5, then x2 = 25. It is ..................... statement. 2 is 5.
x+2=5
f) a + 6 = 9. It is ..................... statement. It is an open
statement.
g) x + 8 = 10, if x = 2. It is ..................... statement.

h) 2x = 6, if x = 4. It is ..................... statement.

2. Let's tell and supply the correct numeral in so that the statement can
be true.

a) The sum of and 6 is 9. The sum of 5 and
9 is 14. It's true
b) The difference of and 5 is 7. statement. And it is
a closed statement.
c) The product of and 4 is 40.

d) The quotient of 42 divided by is 6. e) is greater than 5 by 2.

f) is less than 6 by 3. g) 5 + = 8 h) 10 – = 7

i) – 9 = 0 j) 3 × = 15 k) ÷4=2

3. Let’s tell and write the values of the letters that make the following open
mathematical statements true.

a) x + 4 = 7, x = ............. b) p – 6 = 9, p = .............

c) 5y = 25, y = ............. d) a = 4, a = .............
2
1
e) 3 of x = 6, x = ............. f) A = 7cm × 5cm, A = .............

4. Let’s tell and write the answers as quickly as possible.

a) What should be added to 6 to get 9 ? x + 6 = 9, so, x = ................

b) What should be added to 10 get 16 ? x + 10 = 16, so, x = ................

c) From what should 3 be subtracted to get 5 ? x – 3 = 5, so, x = ................

d) From what should 7 be subtracted to get 3 ? x – 7 = 3, so, x = ................

e) What should be multiplied by 3 to get 6 ? 3 × x = 6, so, x = ................

f) What should be multiplied by 5 to get 20 ? 5 × x = 20, so, x = ................

g) What should be divided by 3 to get 2 ? x = 2 So, x = ................
3
x
h) What should be divided by 5 to get 4 ? 5 = 4, So, x = ................

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5. Let's match each statement with the correct equations.

a) The sum of 3 and a number x is 15 (i) 3x = 15

b) Three less than a number x is 15 (ii) 15 ÷ x = 3

c) The product of three and a number x is 15 (iii) x + 3 = 15

d) 15 divided by a number x is 3. (iv) x – 3 = 15

Creative Section
6. Let's take x as the unknown number and write an equation for each statement.

a) 5 more than a number is 9. b) 4 less than a number is 6.

c) 7 multiplied by a number equals to 21. d) 6 times a number is 30.

e) One-half of a number is 5.

7. A = { 1, 2, 3, 4, 5} is the solution set of the variables in the following

equations. Replace the variables by each element of the set A and state the

solution of each equation. x
2
a) x + 2 = 5 b) y – 3 = 1 c) 3x = 15 d) = 2

8. a) Add 2 to both sides of x – 2 = 3 and find the value of x.

b) Add 5 to both sides of x – 5 = 2 and find the value of x.

c) Subtract 3 from both sides of y + 3 = 7 and find the value of y.

d) Subtract 6 from both sides of y + 6 = 9 and find the value of y.

e) Multiply both sides of a = 3 by 2 and find the value of a.
f) Multiply both sides of a2 = 3 by 5 and find the value of a.
5
g) Divide both sides of 2x = 6 by 2 and find the value of x.

h) Divide both sides of 5x = 10 by 5 and find the value of x.

9. a) What should be added to both sides of x – 3 = 5 to find the value of x ?

Solve it and find the value of x.

b) What should be subtracted from both sides of y + 5 = 9 to find the value

of y ? Solve it and find the value of y.

c) By what should both sides of p = 2 be multiplied to find the value of p ?
3
Solve it and find the value of p.

d) By what should both sides of 5x = 20 be divided to find the value of x ?

Solve it and find the value of x.

It’s puzzle time!
10. Let’s take some matchsticks and make the same patterns as given below.

Then, move only one matchstick to make the equation mathematically
correct.
(i) (ii)

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Equation, Inequality and Graph

10.6 Process of solving equations
While solving equations, we use the equality facts mentioned above.

Worked-out examples

Example 1: Solve x – 4 = 3. Direct method
Solution: x–4 =3
or, x = 3 + 4
x–4=3 –4 is transposed to R.H.S.
or, x – 4 + 4 = 3 + 4 4 is added to both sides. making +4.
or, x = 7 The method is called
transposition method.

Example 2: Solve x + 5 = 8. Direct method
x+5 =8
Solution: or, x = 8 – 5
x+5=8

or, x + 5 – 5 = 8 – 5 5 is subtracted from both sides. +5 is transposed to
or, x = 3 R.H.S. making –5.
The method is called

x transposition method.
3
Example 3: Solve = 4.

Solution: It's my direct method!
x
x =4 3 = 4
3
x or, x = 4 × 3
3
or, u3=4u3 Multiply both sides by 3. Numerator of R.H.S. is
or,
x = 12 multiplied by denominator

of L.H.S.

Example 4: Solve 5x = 15. My methods is called
cross-multiplication.

Solution:

5x = 15 It's my direct method!

5x 15 5x = 15
5 5
or, = Divide both sides by 5. or, x = 15
5
I should divide R.H.S.
or, x = 3
by the coefficient of

Example 5: Solve 6x – 7 = 3x + 5 variable x.

Solution: I can check my answer!
When x = 4.
6x – 7 = 3x + 5 3x is transposed to L.H.S. and 6×4–7=3×4+5
–7 is transposed to R.H.S. or, 24 – 7 = 12 + 5
or, 6x – 3x = 5 + 7 or, 17 =17
So, my answer is correct.
or, 3x = 12

or, x = 12
3
or, x = 4.

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Equation, Inequality and Graph

Example 6: Solve 3x – 1 = 2x + 3 I can check my answer!
5 7
Solution:
When x = 2

3x – 1 = 2x + 3 3×2–1 = 2 × 2 + 3
5 7 5 7

or, 7(3x – 1) = 5(2x + 3) or, 5 = 7 or, 1 = 1
5 7
or, 21x – 7 = 10x + 15 So, my answer is correct.

or, 21x – 10x = 15 + 7

or, 11x = 22

or, x = 11 =2
22

? x = 2.

Example 7: Solve x + x = 3 1 I can also check answer!
2 3 3
Solution:
When x = 4
x x 1 4 4 10
2 + 3 = 3 3 2 + 43 = 130
43 = 130
or, 3x+ 2x = 10 2 + 130
6 3 6 + = 3
310 =
or, 5x = 10 3
6 3
So, the solution is correct.
or, 3 u 5x = 6 u 10

or, 15x = 60
60
or, x = 15 =4

? x =4

EXERCISE 10.2
General Section – Classwork
1. Let’s tell and write the solutions of these equations as quickly as possible.

a) x + 5 = 8, x = ............... b) x + 7 = 10, x = ...............

c) x + 9 = 14. x = ............... d) y – 2 = 4, y = ...............

e) y – 5 = 2, y = ............... f) y – 6 = 6, y = ...............

g) 3a = 9, a = ............... h) 5a = 20, a = ...............
i) 7a = 28, a = ............... p = ...............
j) p = 5,
2 p = ...............

k) p = 4, p = ............... l) p = 6, Vedanta Excel in Mathematics - Book 6
5 9

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Creative Section

2. Solve the following equations and check your answers.

a) x + 4 = 9 b) x + 8 = 15 c) a + 6 = 16

d) a – 5 = 4 e) y – 6 = 2 f) y – 10 = 6

g) 2x + 3 = x + 6 h) 3x – 4 = 2x + 5 i) 9x + 7 = 8x – 3

j) 5x – 8 = 4x – 3 k) 8p – 13 = 7p – 5 l) 5x + 2 = 4x – 7

3. Solve the equations. b) 3x = 12 c) 4y = 24
a) 2x = 6 e) 6m = 54 f) 8k = 56
d) 7y = 42 h) 5x – 3 = 3x + 7 i) 4a + 5 = a + 14
g) 3p – 1 = p + 7 k) 3a + 7 = 7a – 9 l) 2y + 11 = 9x – 10
j) 6x + 11 = 3x + 2

4. Solve the equations.

a) x =2 b) y = 5 c) a = 7 d) 2x =8
5 7 8 3

e) 3p =9 f) 4k = 12 g) a = 3 h) 2x = 232
5 7 4 4 3

i) 4x +3=7 j) 5x – 2 = 8 k) 3a – 1 = 3 1 l) 4x + 2 = 432
5 6 4 2 9

5. Solve the equations.

a) x+1 =2 b) y–3 =2 c) 2a + 3 =3
3 4 5

d) 3x – 1 =2 e) x+1 = 3 f) a–5 = 2
4 4 2 6 3

g) 2x + 1 = x+1 h) 3x – 2 = 2x + 5 i) 2x – 1 = 3x – 2
3 2 4 3 5 7

j) x + x =5 k) 8x – 2x =4 l) 4x – 3x =2
2 3 9 3 5 10

6. From these balances let's make the equations and solve them.

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7. Make an equation for the perimeter of each square. Solve it to find the value
of x. Also find the actual length of each square.

16 cm 24 cm 40 cm

x cm 3x cm (x+2) cm

8. Make an equation for the area of each rectangle. Solve it to find the value of
x. Also find the actual length and breadth of each rectangle.

18 cm2 3 cm 20 cm2 4 cm 36 cm2 (5x–6) cm

(x+2) cm (3x–1) cm 9 cm

9. Make an equation for the perimeter of each rectangle. Solve it to find the
value of x. Also find the actual length and breadth of each rectangle.

20 cm 2x cm 36 cm (x + 2) cm

3x cm (x + 6)

It's your time - Project work!

10. a) Let's write any four equations of the forms x + a = b, x – a = b, ax = b

and x = b, where a and b are integers and x is the variable. Solve your
a
equations and find the value of the variable.

b) Let's write any two equations, in which the solution of one is x = 2 and

the other is y = 3.

10.7 Verbal problems - Use of equations

We use equations to find the unknown value of any quantity. For this, we should
consider the unknown value of the given verbal problems as x, y, z, etc. Then the
verbal problems should be translated into mathematical sentences in the form of
equations. By solving the equations we obtain the required values.

Worked-out examples

Example 1: The sum of two numbers is 15. If one of the numbers is 10, find the
other number.

Solution: Reflection of this question:
There are 15 students in
Let the other number be x. a class. If 10 of them are
Now, x + 10 = 15 boys, find the number of
or, x = 15 – 10 girls.
or, x = 5
Hence, the required number is 5.

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Example 2: If the sum of two consecutive odd numbers is 12, find the numbers.

Solution: Numbers that come one after
another without interruption
Let the smaller odd number be x are called consecutive
Then, the consecutive odd number for x is x + 2.

Now,x + (x + 2) = 12 numbers. 1, 3, 5, 7, 9, ... etc.
are consecutive odd numbers.
or, 2x + 2 = 12 The greater consecutive
number in every pair is 2 more
or, 2x = 12 – 2 than smaller.
So, if smaller odd number is x,
or, x = 10 =5
2

? The smaller odd number = x = 5 the greater will be x + 2.

It’s consecutive odd number = x + 2 = 5 + 2 = 7.

Example 3: If 20% of a sum is Rs 100, find the sum.

Solution: We got the sum = Rs 500

Here,the sum = Rs x Now, 20% of Rs 500

or, 20% of x = Rs 100 = 20 × Rs 500
100
or, 20 ux = Rs 100 or, = Rs 100
or, 100 x = Rs 100
5 which is given in the

or, x = 5 u Rs 100 = Rs 500 question.

Hence, the required sum is Rs 500.

Example 4: If the cost of a dozen of bananas is Rs 30 more than that of
7 bananas, find the cost of a banana.

Solution:

Let the cost of a banana be Rs x.

Then, cost of a dozen of bananas = cost of 12 bananas = Rs 12x.

Now,

12x = 7x + Rs 30 The cost of 1 banana = Rs 6
The cost of 12 bananas = 12 × Rs 6 = Rs 72
or, 12x – 7x = 30 The cost of 7 bananas = 7 × Rs 6 = Rs 42
Now, Rs 72 – Rs 42 = Rs 30 which is given in
or, 5x = 30 the question.
30
or, x = 5 = 6

Thus the cost of a banana is Rs 6.

Example 5: If the sum of two numbers is 10 and the difference is 4, find the
numbers.

Solution:

Let the smaller number be x. Because the difference of a greater and
smaller number is 4 means, the greater
Then, the greater number will be x + 4 is 4 more than the smaller number.

Now, x + (x + 4) = 10
or, 2x + 4 = 10

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or, 2x = 10 – 4
6
or, x = 2 =3

? The smaller number = x = 3.
The greater number = x + 4 = 3 + 4 = 7.

Example 6: The perimeter of a rectangle is 30 cm. If its length is double than
that of its breadth, find the length and breadth of rectangle.

Solution:

Let the breadth of the rectangle be x cm.

Then its length will be 2x cm.

Now, the perimeter of the rectangle = 30 cm

or, 2 (l + b) = 30 cm l = 10 cm and b = 5 cm

or, 2 (2x + x) = 30 cm So, perimeter of rectangle

or, 2 u 3x = 30 cm = 2(l + b)

or, 6x = 30 cm =2(10 + 5) = 30 cm which
is given in the question.
or, x = 30 cm
6

= 5 cm

? The length of the rectangle = 2x = 2 u 5 cm = 10 cm

The breadth of the rectangle = x = 5 cm.

EXERCISE 10.3

General Section – Classwork
1. Let’s tell and write the answers as quickly as possible.

a) If the sum of x and 2 is 7, the value of x is ..............

b) If the difference of x and 4 is 9, the value of x is ..............

c) If the product of x and 3 is 18, the value of x is ..............

d) If the quotient of x divided by 5 is 4, the value of x is ..............

2. Let’s consider the unknown number as x. Tell and write the required
equations as quickly as possible. (Not necessary to solve the equations.)

a) The sum of two numbers is 5 and one of them is 2.

The required equation is ...................................

b) The difference of two numbers is 3 and the smaller one is 6.

The required equation is ...................................

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c) The difference of two numbers is 6 and the greater one is 9.

The required equation is ...................................

d) The product of two numbers is 12 and one of them is 3.

The required equation is ...................................

e) The quotient of dividing a number by 7 is 3.

The required equation is ...................................

Creative Section - A

Make equations and solve them to find unknown numbers.
3. a) The sum of two numbers is 12. If one of them is 7, find the other number.

b) The difference of two numbers is 6. If the greater one is 17, find the
smaller number.

c) The difference of two numbers is 9. If the smaller one is 6, find the
greater number.

d) The quotient of dividing a number by 4 is 5. Find the number.

4. a) Bishwant thought of a number. He doubled it and added 7. If he got 15,
find the number he thought of.

b) Sunayana thought of a number. She trebled it and subtracted 4 from it. If
she got 5, find the number she thought of.

5. a) The cost of 10 pencils is Rs 15 more than the cost of such 7 pencils. What
is the cost of a pencil?

b) The cost of a dozen copies is Rs 50 more than that of 10 copies of same
kind. What is the cost of a copy?

6. a) If the sum of two consecutive odd numbers is 16, find the numbers.
b) If the sum of two consecutive even numbers is 22, find the numbers.

7. a) A sum is Rs x. If its 10% is Rs 40, find the sum.

b) A sum is Rs x. If 2 parts of the sum is Rs 50, find the sum.
3

c) If 25% of a sum is Rs 50, find the sum.

(Hint : 25% of x = Rs 50)

d) 40% of the total number of students of a class are boys. If there are 24
boys, find the total number of students of the class.

8. a) The cost price (C.P.) of an article is Rs x. If 10% profit on C.P. amounts to
Rs 250, find the C.P. of the article.

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b) The cost price (C.P.) of a bag is Rs x. If 15% loss on C.P. amounts to
Rs 120, find the C.P. of the bag.

c) The selling price (S.P.) of a watch is Rs x. If 10% VAT on S.P. amounts to
Rs 180, find the S.P. of the watch.

d) The marked price (M.P.) of an umbrella is Rs x. If 25% discount on M.P.
amounts to Rs 120, find the M.P. of the umbrella.

9. a) The sum of a pair of complementary angles is 90q. If xq and 50q are a pair
of complementary angles, find measure of xq.

b) Two angles are complementary. If one of them is 35°, find the other.

c) The sum of a pair of supplementary angles is 180q. If xq and 80q are a pair
of supplementary angles, find the measure of xq.

d) A pair of angles are supplementary. If one of them is 135q, find the size
of the other angle.

10. Write the equations, solve them and find the size of unknown angles.

(3x+20)°

Creative Section - B

11. a) The perimeter of a rectangle is 18 cm. If its length is 5 cm, find its breadth.

b) The perimeter of a rectangle is 42 cm. If its length is double than that of
its breadth, find the length and breadth of the rectangle.

c) The area of a rectangle is 45 cm2. If its length is 9 cm, find its breadth

d) The area of a rectangle is 54 cm2. It its breadth is 6 cm, find its length.

12. a) If the sum of two numbers is 10 and the difference is 6, find the numbers.

b) A number is greater than another number by 3. If their sum is 33, find the
numbers.

c) A number is less than another number by 7. If their sum is 37, find the
numbers.

13. a) Father is 30 years older than his son. If the sum of their ages is 50 years,
find their ages.

b) Daughter is 25 years younger than her mother. If the sum of their ages is
35 years, find their ages.

14. a) The total cost of a pencil and an eraser is Rs 15. If the pencil is more
expensive than eraser by Rs 4, find their costs.

b) There are 32 students in a class. If the number of girls are 6 more than the
number of boys, find the numbers.

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It's your time - Project work!

15. a) Let's measure the length and breadth of your mathematics book using a
30-cm ruler. By how many centimetres is the length longer than the
breadth? Let's use the answer and do the following problems.

(i) Let the length of the book be x cm. Now, make an equation and find the
value of x. (Hint: x – … = breadth)

(ii) Let the breadth of the book be x cm. Now, make an equation and find the
value of x. (Hint: x + … = length)

b) There are x number of students in your class who like burger and the
rest like pizza. Let's count the number of students who like pizza and
the total number of students. Then, make an equation and find x.

c) There are y number of lady teachers in your school. Let's count the
number of gents teachers and the total number of teachers. Then, make
an equation and find the value of y.

10.8 Trichotomy
Let’s consider any two whole numbers, 4 and 9.

There is only one way to compare these two numbers.

4 < 9 (4 is less than 9) or 9 > 4 (9 is greater than 4).

But, 4 = 9 or 4 > 9 or 9 < 4 are not true comparisons. Thus, if a and b are
any two whole numbers, then only one of the following relations can be true
between them.

Either a = b or a < b or a > b

Such a property of whole numbers is called Trichotomy property. The symbol
‘=’ (equal to), ‘<’ (less than) and ‘>’ (greater than) are called the trichotomy
symbols.

10.9 Negation of trichotomy
Let’s consider any two numbers 2 and 5.

Here, 2 < 5 is true or 5 > 2 is also true.

But, 2 <5 is false or 5 > 2 is also false.
2 < 5 is the negation of 2 < 5 and 5 > 2 is the negation of 5 > 2.

Thus, ‘< ‘ (is not less than) is the negation of trichotomy symbol ‘<’ (less than)

‘> ‘ (is not greater than) is the negation of trichotomy symbol ‘>’ (greater than)

‘z' (is not equal to) is the negation of trichotomy symbol '=' (equal to).

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10.10 Trichotomy rules
Let’s consider any two numbers 6 and 8.

Rule 1: I got it!
6<8
6+2<8+2 or 8 > 6 is true If x < a or x > a, then
i.e. 8 < 10 or 8 + 2 > 6 + 2
or 10 > 8 is also true. x + b < a + b or x + b > a + b

Where, a and b are natural numbers

Thus, when an equal number is added to both sides of trichotomy symbol, the

symbol remains the same.

Rule 2: I also understood!

6<8 or 8 > 6 is true If x < a or x > a, then
6 – 2<8 – 2 or 8 – 2 > 6 – 2 x – b < a – b or x – b > a – b
i.e. 4 < 6
or 6 > 4 is also true. Where, a and b are natural numbers

Thus, when an equal number is subtracted from both sides of trichotomy

symbol, the symbol remains the same.

Rule 3: Now, it's easier!

6<8 or 8 > 6 is true If x < a or x > a, then

2 u 6 < 2 u 8 or 2 u 8>2 u 6 b × x < b × a or b × x > b × a

i.e. 12 < 16 or 16 > 12 is also true where, a and b are natural numbers.

Thus, when both sides of trichotomy symbol are multiplied by an equal
number, the symbol remains the same.

Rule 4: I also investigated!

6<8 or 8 > 6 is true If x < a or x > a, then
6 8 8 6 x a x a
2 < 2 or 2 > 2 b < b – b or b > b

i.e. 3 < 4 or 4 > 3 are also true. where a and b are natural numbers

Thus, when both sides of trichotomy symbol are divided by an equal positive
number, the symbol remains the same.

It's interesting!
Rule 5: x a
If x < a, then –b > –b
6<8 or 8 > 6 is true
6 8 6 6 x < a, then x < a
–2 < –2 or –2 > –2 are not true. –b –b

i.e. – 3 < – 4 or –4 > – 3 are not true. where a is positive and b is

But, – 3 > – 4 or – 4 < –3 are true. negative integers.

Thus, when both sides of trichotomy symbol are divided by an equal negative
number, the symbol ‘<’ is changed to ‘>’ or ‘>’ is changed to ‘<’.

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Worked-out examples

Example 1: Insert the appropriate trichotomy symbols (<, > or =) in the

boxes. 2 b) – 4 –9+5 c) 6 8
a) – 3 –2 –2

Solution: 6 8
–2 –2
a) – 3 < 2 b) – 4 = – 9 + 5 c) >

Example 2: Rewrite the statement ‘– 2’ is less than x’ using trichotomy symbol.
Also write it by using negation symbol.

Solution:

– 2 is less than x

– 2 < x (using trichotomy symbol)

– 2 < x (using negation symbol)

EXERCISE 10.4
General Section – Classwork

1. In the patterns below, each side of 6 < 12 is increased, decreased,
multiplied or divided by the same non-zero number. Let’s compare the
values, investigate the idea and insert < or > sign in the box.

Addition pattern Subtraction pattern

6 < 12 6 < 12

6+3 12 + 3 6–3 12 – 3

6+2 12 + 2 6–2 12 – 2

6+1 12 + 1 6–1 12 – 1

6 + (–1) 12 + (–1) 6 – (–1) 12 – (–1)

6 + (–2) 12 + (–2) 6 – (–2) 12 – (–2)

6 + (–3) 12 + (–3) 6 – (–3) 12 – (–3)

Multiplication pattern Division pattern

6 < 12 6 < 12

6×3 12 × 3 6÷3 12 ÷ 3

6×2 12 × 2 6÷2 12 ÷ 2

6×1 12 × 1 6÷1 12 ÷ 1

6 × (–1) 12 × (–1) 6 ÷ (–1) 12 ÷ (–1)

6 × (–2) 12 × (–2) 6 ÷ (–2) 12 ÷ (–2)

6 × (–3) 12 × (–3) 6 ÷ (–3) 12 ÷ (–3)

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2. Let’s tell and write ‘true’ or ‘false’ in the blank spaces.

a) – 3 > 0 ............ b) – 2 < – 3 .......... c) – 4 > – 7 ...............

d) 4 > – 7 ........... e) – 2 + 3 < – 3 + 2 ............. f) 9 – 6 < 6 – 9 ...........

g) 1 × (– 6) < (– 3) × 2 ............ h) 10 = – 10 ......... i) – 21 > 21 ............
–5 5 –7 –7

3. Let's insert the appropriate trichotomy symbols (<, > or =) in the blank

spaces.

a) 8 ....... 5 b) 7 ....... 11 c) – 2 ....... – 5 d) – 4 ....... 2

e) 0 ....... 6 f) – 3 ....... 2 – 5 g) (–2) × (– 6) ....... – 12 h) 18 ....... 12
–9 –6

4. Rewrite the following trichotomy statements using negation symbols.

a) x < 4 ....................... b) y > 2 ....................... c) x + 5 = 8 .......................
Creative Section

5. Write the mathematical statement corresponding to each statements by

using the appropriate sign (< or > or =).

a) x is greater than 6 b) y is less than 2

c) x is equal to –4 d) m is a positive number

e) p is a negative number f) k is neither positive nor negative number

6. Rewrite the following statements using trichotomy symbols. Also write
the negation of each statement and represent them by using negation of
trichotomy symbols.

a) x is greater than 5 b) y is less than – 3
c) z is equal to 6 d) – p is greater than – 2
e) – x is less than – 4 f) x + 5 is equal to 9
g) y – 8 is greater than – 2 h) 7 – x is less than x – 7

10.11 Inequalities

We have already discussed that an equation is the open statement containing
‘equal to’ (=) sign. On the other hand, if an open statement contains the symbol
of ‘less than (<)’, ‘greater than (>)’, ‘less than and equal to ( d ) or ‘greater than and
equal to (t)’, it is called an inequation. An inequation is also called inequality.
For example:

x is less than 4 is x < 4,
y is greater than 7 is y > 7,
(x + 1) is less than and equal to 6 is x + 1 d 6,
(y – 5) is greater than and equal to 9 is y – 5 t 9, etc. are a few examples of
inequalities.

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10.12 Solution of inequalities
Let’s consider an inequality x + 2 < 6.
This inequality can be true for more than one value of x. For example:

When x = – 1, – 1 + 2 < 6 or 1 < 6, it is true.
When x = 0, 0 + 2 < 6 or 2 < 6, it is true.
When x = 1, 1 + 2 < 6 or 3 < 6, it is true.
When x = 2, 2 + 2 < 6 or 4 < 6, it is true.
When x = 3, 3 + 2 < 6 or 5 < 6, it is true.
When x = 4, 4 + 2 < 6 or 6 < 6, it is not true.
When x = 5, 5 + 2 < 6 or 7 < 6, it is not true.

Thus, the inequality x + 2 < 6 is true for many values of x. The set of different
values of x that make the inequality true is called the solution set of the inequality.

Here, the solution set of x + 2 < 7 is {…– 1, 0, 1, 2, 3}.

We should use the trichotomy rules to find the solution sets of inequalities.

Worked-out examples

Example 1: Represent the following inequalities on number lines.

a) x > 2 b) x d 3 c) –1 < x d 4
Solution:

a) x > 2 b) x d 3

Note: The empty circle at 2 shows that Note: The solid circle at 3 shows that
2 is not included in the solution set. 3 is included in the solution set.

c) – 1 < x d 4

Example 2: A = {1, 2, 3, 4, 5} is the replacement set. Replace the variable of
the inequality x + 1 > 3 by each element of the replacement set
and find the solution set.

Solution:

When x = 1, 1 + 1 > 3 or 2 > 3, it is not true.

When x = 2, 2 + 1 > 3 or 3 > 3, it is not true.

When x = 3, 3 + 1 > 3 or 4 > 3, it is true.

When x = 4, 4 + 1 > 3 or 5 > 3, it is true.

When x = 5, 5 + 1 > 3 or 6 > 3, it is true.

? The solution set = {3, 4, 5}.

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Example 3: Find the solution sets of the following inequalities and represent

them on number lines.

a) x + 2 < 4 b) x – 3 > 5 c) 2x + 3 t 9

Solution: x<2

a) Here, x + 2 < 4

or, x + 2 – 2 < 4 – 2

or, x < 2

? Solution set = {…. – 1, 0, 1}

b) x – 3 > 5

or, x – 3 + 3 > 5 + 3

or, x > 8
? Solution set = {9, 10, …}

c) 2x + 3 t 9

or, 2x + 3 – 3 t 9 – 3

or, 2x t 6 6
2x 2
or, 2 t

or, x t 3

? Solution set = {3, 4, 5,…}

EXERCISE 10.5
General Section – Classwork

1. Let’s make the mathematical statements by using inequality symbols.

a) x is greater than 4 ....................................................

b) y is less than 7 ....................................................

c) x is greater than or equal to 3 ....................................................

d) y is less than or equal to –2 ....................................................

e) x – 1 is greater than or equal to 3 ....................................................

f) x + 5 is less than and equal to - 4 ....................................................

2. Let’s tell and write the solution sets as quickly as possible.

Eg. x < 3 {2, 1, 0, – 1, –2, ...} x > – 2 {– 1, 0, 1, 2, 3, ...}

a) x < 5 ......................................................................................

b) x > 2 ......................................................................................

c) x > – 4 ......................................................................................

d) x < – 3 ......................................................................................

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3. Write the inequalities represented by these number lines.

Creative Section

4. A = {1, 2, 3, 4, 5} is the replacement set. Replace the variables of the
inequalities by each element of the replacement set and find the solution
set.

a) x + 2 < 7 b) x + 4 > 5 c) x – 3 d d) x – 1 t 1

5. Represent the following inequalities on number lines.

a) x < 5 b) x < – 3 c) x d d) x t – 2

6. Find the solution sets of the following inequalities and represent them on
number lines.

a) x + 3 < 7 b) x + 4 > 6 c) x – 2 d 4 d) x – 5 t – 4

7. Write the inequalities represented by the following number lines. Also state
the solution sets.

e) f)

8. Let's answer the following questions.

a) x represents the number of members of a club. At least 15 members are
needed to conduct the weekly club meeting. Represent this statement
using inequality symbol.

b) The length of a square is x cm. Write an inequality to show it's perimeter
at least 20 cm. Solve the inequality and find the minimum length of the
square.

c) x represent the average quantity of rice to be sold everyday to clear the
stock of 350 kg of rice in a week. Represent it in inequality and find the
average quantity of rice to be sold everyday.

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Unit Coordinates

11

11.1 Ordered pairs - Looking back
Classwork - Exercise

Let’s tell and write the answers as quickly as possible.
1. a) Write any three ordered, pairs such that the first member is an animal

and the second member is it's baby animal.
................................... ................................... ....................................
b) Write any four ordered pairs such that the first member is a square num-
ber and the second member is it's square root.
...................... ...................... ...................... ......................
2. a) If the coordinates of a point is (4, 6), the x - coordinate is ........... and the
y - coordinate is ...............
b) If x - coordinate of a point is 5 and y - coordinate is 2, the coordinates of
the point are ..............

c) The coordinates of the origin are ........................

3. Let’s look at the graph. Tell and write the coordinates of the points as
quickly as possible.

10
9

8 Coordinates of the point P are ...................
Coordinates of the point Q are ...................
7 Coordinates of the point R are ...................
Coordinates of the point S are ...................
6 R

5 Q
4 P

3

2

1S
0 1 2 3 4 5 6 7 8 9 10

Look at a few countries and their capitals. These Nepal Kathmandu
India New Delhi
countries and their capitals can be written in China Bejing
pairs. Bhutan Thimpu

(Nepal, Kathmandu), (India, New Delhi),

(China, Bejing), (Bhutan, Thimpu), Manmar Naypyidaw

(Myanmar, Naypyidaw).

These are the special pairs in which the first member is always country and

the second member is its capital. So, the countries and capitals are in the fixed

order. Such pairs are called ordered pairs.

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Coordinates

11.2 Coordinates
In the figure alongside, two number
lines OX and OY are perpendicular
to each other at the point O. Here, the
horizontal number line OX is called
x–axis and the vertical number line
OY is called y–axis. O is called the
origin.

Now, we can find the position of
each point in terms of its ordered
pair in which the first member is
the number of squared rooms along OX (x–axis) and the second member is the
number of squared rooms along OY (y–axis). For example:

The position of the point P is (5, 3). So, the coordinates of the point P are (5, 3).

Here, 5 is called the x–coordinate or abscissa and 3 is the y–coordinate or
ordinate.

Similarly, the coordinates of the point Q are (6, 8). The x–coordinate 6 comes
first, then the y–coordinates 8; along, then up.

The coordinates system which describes the accurate position of points was
invented by a Frenchman called Rene’ Descartes in the 17 the century.

In the adjoining graph, OX' and OY'
are negative number lines. Negative
x–coordinates are plotted along OX' and negative
y–coordinates are plotted along OY'.

For example,
The coordinates of the point A is (– 2, 3)
The coordinates of the point B is (– 3, – 2)

Note: (i) The coordinates of the origin O is (0, 0)

(ii) The coordinates of a point on x–axis is (x, 0), where x is the number of
squared rooms counting from origin along x–axis.

(iii) The coordinates of a point on y–axis is (0, y), where y is the number of
squared rooms counting from origin along y–axis.

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Coordinates

EXERCISE 11.1

General Section – Classwork
1. Let's tell and write the correct answers as quickly as possible.

a) If the coordinates of a point is (4, 7), the x-coordinate is .................... and
the y-coordinate is .............................

b) If the x-coordinate of a point is 5 and y-coordinate is 3, the coordinates
of the point are .............................

c) The coordinates of the origin are .............................

d) A point is 2 units right along x-axis and 6 units up along y-axis from the
origin. It’s coordinates are .................................

e) A point is 3 units left along x-axis and 2 unit up along y-axis from the
origin. It’s coordinates are .............................

f) The coordinates of a point on x-axis 5 units right from the origin
are .............................

g) The coordinates of a point on y-axis 4 units up from the origin

are .............................

2. Let's tell and write the coordinates of the points from the graph as quickly
as possible.

Coordinates of the point A are (.................)

G Coordinates of the point B are (...............)
A Coordinates of the point C are ...............

Coordinates of the point D are .................
F E Coordinates of the point E are .................

Coordinates of the point F are .................

Coordinates of the point G are .................

H Coordinates of the point H are .................

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Coordinates

3. Plot these points in the graph as quickly
as possible.

A (2, 1), B (– 3, 4)

C (– 4, – 1), D (–1, 3)

E (3, 0), F (0, 2)

G (–2, 0), H (0, –4)

4. Answer the following questions.

a) What is the y–coordinate of a point on the x–axis? ................

b) What is the x–coordinate of a point on the y–axis? ................

c) On what axis does the point (4, 0) lie? What is the ordinate of this
point? ................

d) On what axis does the point (0, – 3) lie? What is the abscissa of this
point? ................

Creative Section - A
5. Write down the coordinates of the vertices of these shapes.

Hint: In PQRS, P (– 3, 7), Q (0, 7), R (– 3, 4), S (– 6, 4)

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Coordinates

6. On a squared paper, draw XOX' and YOY' axes which are mutually
perpendicular at O (origin). Then plot these points.

A (2, 3), B (5, 4), C (– 3, 5), D (– 6, 2), E (5, 0), F (0, 2)

G (– 1, – 5), H (0, – 6) P (5, – 2), Q (– 7, 2), R (– 4, – 2), S (– 4, – 6)

7. Plot the following points and join them in alphabetical order by using ruler.
Name the figures so formed.

a) A (3, 2), B (5, 3), C (1, 5)

b) P (1, 1), Q (3, 1), R (3, 3), S (1, 3)

c) M (3, 1), A (3, 4), T (–2, 1), H (–2, 4)

d) L (–1, –1), I(3, –1), K(3, 2), E(–1, 2)

e) N (– 2, 2), I (0, 5), C (2, 2) E (0, – 7)

f) B (– 4, 2), E (– 7, – 1), S (– 4, – 4), T (– 1, – 1)

g) Y (1, – 2), E (3, 0), A (3, 3), R (– 1, 3)
Creative Section - B

8. a) Plot the points A(1, 2) and B(5, 2) on a graph paper and join them. Mark
the mid-point M of line segment AB and find its coordinates.

b) Plot the points P(2, 3) and Q(6, 5) on a squared paper and join them.
Mark the mid-point N on the line segment PQ and write its coordinates.

9. a) Draw a line segment AB with ends A(–1, 3) and B(2, 3). Produce AB to the
point C so that AB = BC. Then, write the coordinates of the point C.

b) Draw a line segment LM with ends L(–2, –1) and M(–2, 1). Extend LM to
the point N. So that LM = MN. Write the coordinates of point N.

It's your time - Project work!

10. a) Let's mark a points respectively in the first, second, third and the fourth
quadrants of the coordinates plane in a graph paper. Then, write the
coordinates of the point in each quadrant.

b) Let's draw a rectangle in a graph paper. Write the coordinates of the
vertices of the rectangle.

c) Let's draw a square in a graph paper write the coordinates of the vertices
of the square .What is the area of your square?

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Unit Geometry: Point and Line

12

12.1 Point, line, ray, line segment and plane - review

Geometry is a branch of mathematics dealing with shape, size and other
properties of different figures. The word ‘geometry’ is derived from the Greek
word ‘geometron’ where ‘geo’ means ‘Earth’ and ‘metron’ means measurement.

In geometry, we study about point, line ray, line segment, plane, angles different
plane shapes, solid shapes, geometrical constructions, etc.

Point AB

It is a mark of a position. R C
It has no length, breadth or height.

It is denoted by capital letters.

In the given figure, P, Q, R, A, B, C, etc. are different points.

Real life examples: A dot made by the tip of a sharp pencil, location of places on

a map, stars in the sky, thumbtack, etc.

Line
It is a straight path which can be extended
indefinitely in both the directions.
It is shown by two arrow heads in opposite
directions.
It can be straight or curved but it is a straight line
when we simply say 'a line'.
It can never be measured because it has no
endpoints.

In the figure, AB and PQ are straight lines. XY is a curved line.
Real life examples: Number line, rubber band while stretching both directions, etc.

Note:
(i) An infinite number of lines can be drawn through a given points.
(ii) One and only one straight line can be drawn through two points.
(iii) An infinite number of curved can be drawn through two points.

Ray

It is a straight path which can be extended indefinitely

only in one direction and the other end is fixed. R
Q
It's fixed end is also called the initial point.
P
The end which can be extended is shown by an

arrowhead. O

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Geometry – Point and Line

In the figure, AB, PQ, OQ and QR are rays.
Real life examples: sun rays, ray of light of torchlight, projector, etc.

Line segment
A line segment is a part of a straight line, It has a
definite length. In the figure, AB and PQ are line
segments. Real life examples: a piece of wire,
pencil, ribbon, edge of book, table, etc.

Plane
It is a flat surface. A plane has length and width, but no thickness.
In the figure, there are six plane surfaces of the cube. Triangles,
quadrilateral, pentagon, etc. are some examples of plane figure.

12.2 Intersecting line segments
Let's study the following illustrations.

a) Two roads meeting at a place making cross roads.

b) Two arms of scissors meet at a point.

c) A The minute hand OA and the hour hand OB of a watch meet each
other at a point O.
O
B

In each of the above figures, the line segments cross each other at a point.

Thus, two or more line segments are said to be the intersecting line segments if
they cross each other through a common point.

In the figure (i), line segments AB and CD
cross each other through a common point O.
So, AB and CD are intersecting line segments.
Similarly, in figure (ii), AB, CD and EF are
intersecting line segments.

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Geometry – Point and Line

12.3 Parallel line segments AP S Q
D R
Let's observe the opposite edges of your
textbook, table, desk, whiteboard, etc.

and investigate idea about the parallel B

lines. C
Two or more line segments are said to

be parallel if the perpendicular distance

between them is always equal. Parallel line segments do

not meet each other when they are produced to either

directions.

In the figure, AB and CD are two parallel line segments
because their perpendicular distances PQ and RS are
equal.

Parallel line segments are represented as and AB
parallel to CD is written as AB//CD.

Note that, if AB // CD and CD // EF then AB // EF.

12.4 Perpendicular line segments 90°
90°
Let's observe angle between each stair, adjacent edge of your
exercise book, angle between the hands at 9 o'clock in a watch, etc. stairs
and investigate the idea of perpendicular lines.

Two line segments are said to be
perpendicular if they meet (or intersect)
each other making an angles of 90q. In
the figure (i), CD is perpendicular to AB
at D. We write it as CD A AB. Similarly,
in the figure (ii) RS A PQ at O.

Note that, if AB A MN and CD A MN
then AB A CD.

EXERCISE 12.1

General Section – Classwork
1. Let's read the incomplete sentences given below and write 'parallel' or

'perpendicular' to complete them.

a) The opposite edges of a ruler are .....................

b) The arrangements of books in the bookshelf are .....................

c) The X-axis and Y-axis are .....................

d) The adjacent sides of a rectangle are .....................

e) The diagonals of a square are .....................

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Geometry – Point and Line

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

a) How many straight lines can be drawn from a point ? ................

b) How many straight lines can be drawn through two points ? ................

c) How many curved lines can be drawn through two points ? ................

d) Through how many points do two lines interest each other ? ..................

e) What is the angle made by two perpendicular lines ? ..................

f) What is the angle made by two parallel lines ? ..................

g) If the perpendicular distances between two lines at any

point are equal, the lines are said to be ..................

h) Are the opposite edges of your desk parallel ? ..................

i) Are the breadths of your books perpendicular to their lengths ? ...............

j) How many plane surfaces are there in your exercise book ? ..................

3. Look at the adjoining figure. Tell and write the pairs of D C
parallel and perpendicular lines in geometrical notations.
O
Parallel lines are : (Hint : AB// CD) ............................. A PB
Perpendiculars are : ....................................................

4. a) If AB//CD and CD//EF, what is the relation between AB and EF?

........................................................................
b) If PQ A AB and RS A AB, what is the

relation between PQ and RS?

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

c) If KL A PQ and MN A KL, what is the relation
between MN and PQ?

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

d) If AB A XY and CD A AB, what is the relation
between XY and CD?

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

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Geometry – Point and Line

e) If AB // CD, EF and GH are perpendicular lines A E GB
on AB and CD, what is the relation between EF F HD
and GH?

................................................................ C

f) If WX = YZ, what is the relation between AB C
and CD?

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

g) How many perpendiculars can be drawn from
the point P on the line AB?

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

h) How many lines parallel to PQ can be drawn
through the point A?

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

i) If KL is perpendicular to XY at L but it is
not perpendicular to PQ. What do you say
about XY and PQ?

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

Creative Section

5. Name the points that represent the vertices of these figures.

a) A b) S R c) D
C
E

B C Q
P
AB

6. Draw these straight lines, curved lines, rays, line segment in separate

groups. Also write their names. Y NF

P QR S

CD

X

ME

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Geometry – Point and Line

7. Name the line segments and their points of intersection. X
P
a) A C b) R T M c)

K L

O P Q MQ N
A

DB NU S Y

8. Name the parallel and perpendicular line segments. Also represent them in
geometrical notation.

E.g. AB parallel to CD (AB//CD). c) d)
AB perpendicular to CD (AB A CD)
a)
b)

9. Name the parallel and perpendicular line segments. Express them in
geometrical notations.

a) b) c)

10. Copy the tables and write the measurements of the perpendicular distance
between each pair of line segments. State whether the line segments are
parallel or not.

PQ RS TU KL MN OP AB CD EF

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Unit Geometry: Angle

13

13.1 Angles (review) arms
vertex
A corner made by meeting two straight line segments is known as
an angle. In the figure ‘AOB (or ‘BOA or ‘O) is an angle formed
by the line segments AO and BO meeting at a corner point O.
Here, AO and BO are called the arms and O is called the vertex of
‘AOB.

13.2 Angles formed by a revolving line

In the following figures, OX be the fixed line segment. It is also called the initial
line segment. OP be the revolving line segment that turns about a fixed point O
in anticlockwise direction.

A quarter turn, 90q Half turn, 180q Third quarter turn, 270q A complete turn, 360q

13.3 Measurement of angles

To measure angles, we use protractor in the following steps.
Step 1: Place the centre of protractor on the vertex of the angle.
Step 2: Line up one of the arms of the angle with the base line (zero line) of the

protractor.
Step 3: Count up the angles in degrees starting from 0° until the other arm meets

the protractor.
Let’s observe carefully and learn to measure angles by using a protractor.

The arm along the base line passes through 0° on The arm along the base line passes through 0° on
the outer scale. So, outer scale is used here. the inner scale. So, inner scale is used here.

Outer scale B

Inner scale

Use outside scale ‘POQ = 75q A

Use inside scale ‘AOB = 120q

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Geometry – Angles

13.4 Types of angles
According to the sizes of angles, they are categorised into different types, such
as acute angles, right angle, obtuse angles, straight angle, reflex angles and an
angle of a complete turn.

An acute angle, A right angle, 90q An obtuse angle,
between 0q and 90q 90q between 90q and 180q

180q 90q

180q 0q

A straight angle, 180q A reflex angle, 270q

between 180q and 360q An angle of a
complete turn, 360q

Now, let’s remember these types of angles and their sizes.

Types of angles Sizes Examples
1. Acute angles
2. Right angle Greater than 0q and less than 90q 10q, 24, 55, 89q, etc.
3. Obtuse angles
Exactly 90q 90q
4. Straight angle
5. Reflex angles Greater than 90q and less than 98q, 120q, 135q, 160q, etc.

6. Angles of a 180q
complete turn
Exactly 180q 180q

Greater than 180q and less than 181q, 210q, 270q, 300q,

360q etc.

Exactly 360q 360q

EXERCISE 13.1
General Section – Classwork

1. Let's tell and write the names, vertices and arms of these angles.

Name ............... Name ...............

Vertex ............... Vertex ...............

Arms ............... Arms ...............

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Geometry – Angles

2. Let's tell and write the names and sizes of these angles.

Name ............... Name ...............
Size ............... Size ...............

Name ...............
Size ...............

Name ...............
Size ...............

3. Let's use protractor to measure the sizes of these angles and write them
with their names.

Name ............... Name ...............
Size ............... Sizes ...............

4. Let's tell and write below whether these angles are acute, right, obtuse or
reflex angles.

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

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Geometry – Angles

5. Let's list separately these angles as acute, obtuse, right, straight or reflex
angles.
80°, 120°, 50°, 180°, 210°, 90°, 330°, 360°, 150°, 270°.

Acute Right Obtuse Straight Reflex

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

a) The quarter turn of a revolving line makes an angle of ..................

b) The half turn of a revolving line makes an angle of ..................

c) The third - quarter turn of a revolving line makes an angle of ..................

d) The complete turn of a revolving line makes an angle of ..................

e) The angle made by a circle is ..................

f) The degree measurement of 1 right angle is ..................

g) The degree measurement of 2 right angle is ..................

h) If x° and 40° make a right angle, then x° = ..................

i) If y° and 120° make a straight angle, then y° = ..................

j) If z° and 300° make an angle of complete turn, then z° = ..................

7. Let's look at the given watch. Tell and write in how many minutes will the
revolving minute hand turn through.

a) a quarter turn .................... minutes

b) a half turn .......... minutes.

c) a third-quarter turn ........ minutes

d) a complete turn .... minutes.

Creative Section

8. Make equations and solve them to find unknown angles.
a) If x° and 60° make a right angle, find the size of x°. (Hint : x° + 60° = 90°)
b) If y° and 100° make a straight angle, find the size of y°.
c) If z° and 240° make an angle of a complete turn, find the size of a°.
d) If a° and 75° make an angle of quarter turn, find the size of a°.
e) If p° and 115° make an angle of half turn, find the size of p°.

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Geometry – Angles

9. A revolving minute hand of a watch subtend an angle of 360q in 60 minutes
(1 hour). Apply unitary method and calculate the following.
a) The angle subtended by the minute hand in 10 minutes.
b) The angle subtended by the minute hand in 25 minutes.
c) The time taken by the minute hand to turn through 90q.
d) The time taken by the minute hand to turn through 210q.

10. Calculate the size of unknown angles.

40° 25° 110°

30°
Hint: x° + 30° = 90°

150° 4x°
2x°

3x°

13.5 Different pairs of angles

There are a few special pairs of angles. They are adjacent angles, liner pair,
vertically opposite angles, complementary angles and supplementary angles.

(i) Adjacent angles

Two angles are said to be adjacent angles if they have
a common vertex and a common arm. In this figure,
‘AOB and ‘BOC are adjacent angles because they have
a common vertex O and a common arm OB.

(ii) Linear pair

Two angles are said to be a linear pair if their sum is
a straight angle (180°). In the figure, ‘AOB and ‘BOC
are linear pair because they are the adjacent angles
whose sum is a straight angle (180q).

? ‘AOB + ‘BOC = straight angle = 180q

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Geometry – Angles

(iii) Vertically opposite angles

When two line segments intersect at a point then a pair
of angles lying opposite side of common vertex are called
vertically opposite angles. In the figure, ‘AOC and
‘BOD are vertically opposite angles. Vertically opposite angles are equal.
? ‘AOC = ‘BOD and ‘AOD = ‘BOC

(iv) Complementary angles
Two angles are said to be complementary if their sum is 90°. In the given
figure, ‘AOB + ‘BOC = 90°.
Therefore, ‘AOB and ‘BOC are complementary angles.
Also, complement of ‘AOB = 90q – ‘BOC and complement
of ‘BOC = 90q – ‘AOB.

(v) Supplementary angles
Two angles are said to be supplementary if their sum is 180°. In the given
figure ‘AOB + ‘BOC = 180q
Therefore, ‘AOB + ‘BOC are supplementary angles.
Also, supplement of ‘AOB = 180q – ‘BOC and supplement
of ‘BOC = 180q – ‘AOB

Worked-out examples

Example 1: Find the supplement of 110°.
Solution:
Here, the supplement of 110q = 180q – 110q = 70q

Example 2: If 4xq and 5xq are a pair of complementary angles, find them.

Solution:

Here, 4xq + 5xq = 90q [The sum of a pair of complementary angle is 90q]
or, 9xq = 90q

or, xq = 90° = 10q
9

? 4xq = 4 u 10q = 40q and 5xq = 5 u 10q = 50q.

Example 3: If xq and (x + 20)q are adjacent angles in linear pair, find them.

Solution:

Here,x° + (x + 20)° = 180° [Being linear pair]

or, 2xq = 180q – 20q
160°
or, xq = 2 = 80q

?xq = 80q and (x + 20)q = (80 + 20)q = 100q.

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Geometry – Angles

Example 4: In the figure below, find the sizes of unknown angles.

a) b) c)

5x° 60° 2x° 125° y°
3x° (x + 12)° z° x°

Solution:

a) Here, 5x° + 60° = 180° [Being linear pair]

or, 5x° = 180° – 60°

or, 5x° = 120°
120°
or, x = 5 = 24°

? xq = 24° and 5x = 5 × 24° = 120°

b) Here,(x + 12)° + 2x° + 3x°= 180q [Sum of the parts of a straight angle (180q)]

or, 6x° + 12° = 180q

or, 6xq = 180q – 12q
168°
or, xq = 6 = 28q

? (x + 12)° = 28° + 12° = 40°, 2x° = 2 × 28 = 56° and 3x° = 3 × 28° = 84°

c) Here, (i) xq = 125q [Being vertically opposite angles]

(ii) x° + yq = 180q [Being linear pair]

or, 125° + y° = 180° [The sum is a straight angle (180q)]

or, yq = 180° – 125° = 55°

(iii) z° = y° = 55° [ Being vertically opposite angles]

? x = 125°, y° = z° = 55°

EXERCISE 13.2
General Section – Classwork

1. Let's look at the figure alongside, tell and write the answers in the blanks

as quickly as possible. E

a) ∠COE and ................. are a pair of adjacent angles. C

b) ∠BOC and ................ are vertically opposite angles.

c) ∠BOD and ∠AOC are ................................. angles. A OB

d) ∠BOC is the complement of ......................... D

e) ∠ AOD and ....................... are adjacent angles in linear pair.

f) The supplement of ∠AOC is ..................

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Geometry – Angles

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

a) The complement of 60° is ........................

b) The supplement of 150° is ........................

c) If x° and 110° are linear pair, then x° = ........................

d) If y° and 85° are vertically opposite angles, then y° = .......................

Creative Section - A

3. State with reason whether ‘a and ‘b are adjacent angles or not in the

following figures.

a) b) R c) Y

Q

ba S ba PZ ab X
O O

d) A e) L f) M

aB Jb X
Cb
a Kb
D M
N Ia S

4. From the figure, state with reason whether ‘x and ‘y are vertically

opposite angles are not.

a) S b) C c)A F dL) R

P x QA xy BB x EM yO Q
yO Oy D x P

R D C N

5. Copy the figures and name the pairs of adjacent angles and vertically
opposite angles separately.

O

6. a) Find the complements of (i) 55° (ii) 20° (iii) 62° (iv) 30°

(Hint : complement of 50° = 90°– 50° = 40°)

b) Find the supplements of (i) 45° (ii) 110° (iii) 150° (iv) 60°

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Geometry – Angles

7. Make equations and solve them to find unknown angles.
a) If x° and 125° are adjacent angles in linear pair, find x°.

(Hint : x° + 125° = 180°)

b) If 2x° and 3x° are adjacent angles in linear pair, find them.
c) If y° and (y + 10)° are complementary angles, find them.
d) If (a + 10)° and (a + 20°) are supplementary angles, find them.
e) If two complementary angles are equal, find them.
f) If two supplementary angles are equal, find them.
8. Find the sizes of unknown angles.

100° 105° 3y°
55° 2y°

z° 68°
4z°

Creative Section - B
9. a) Find the supplement of complement of 50°.

b) Find the complement of supplement of 100°.

10. Find the sizes of unknown angles.

a) b) c) d)

11. Find the sizes of unknown angles.

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Geometry – Angles

12. a) Copy the diagram and calculate the sizes of xq,
yq and zq. What is the sum of the angles of the
triangle?

b) Copy the diagram and calculate the sizes of
aq, bq and cq. What is the sum of the angles of
the triangle?

It's your time - Quiz time!
13. Tick (√) the correct option.

a) ...................... angles have a common vertex and a common arm.

(i) adjacent angle (ii) vertically opposite angles

(iii) complementary angles (iv) supplementary angles

b) Vertically opposite angles are always ......................

(i) supplementary (ii) complementary

(iii) equal (iv) unequal

c) Two right angles always form a ...................... pair.

(i) straight angle (ii) linear pair

(iii) supplementary angles (iv) all of the above

d) The component of 1 of 180° is ......................
3

(i) 30° (ii) 45° (iii) 60° (iv) 90°

e) The supplement of 1 of 90° is ......................
2

(i) 45° (ii) 135° (iii) 60° (iv) 90°

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 197 Vedanta Excel in Mathematics - Book 6

Geometry – Angles

15.6 Angles made by a transversal with straight line segments

In the given figure, PQ is a straight line segment that

intersects two straight line segments AB and CD at the E
points E and F respectively. Here, PQ is called a transversal.

In this way, different pairs of angles are formed. F

(i) ‘AEF and ‘CFE are a pair of co-interior angles.
(ii) ‘AEF and ‘EFD are a pair of alternate angles.
(iii) ‘AEP and ‘CFE are a pair of corresponding angles.

13.7 Pairs of angles made by a transversal with parallel line
segments

In the adjoining figure, AB and CD are two parallel
line segments. Transversal PQ intersects AB at S and
CD at R.

Let's learn about the properties of the following pairs
of angles made by a transversal with parallel line
segments.

(i) Interior angles
‘a, ‘b, ‘c and ‘d are lying inside the parallel
lines. They are called interior angles.

Consecutive interior (or co–interior) angles:
‘a and ‘c are a pair of co–interior angles. They
are lying to the same side of the transversal.
Similarly, ‘b and ‘d are another pair of
co–interior angles.

The sum of a pair of co–interior angles made by a transversal with parallel

lines is always 180q.

? ‘a + ‘c = 180q and ‘b + ‘d = 180q

‘a and ‘b are co–interior angles. So, ‘a + ‘b = 180q.

Vedanta Excel in Mathematics - Book 6 198 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Geometry – Angles

(ii) Alternate interior angles (or alternate angles)
‘a and ‘d are a pair of alternate interior angles.
They are simply called alternate angles. They are
lying to the opposite side of the transversal and they
do not have a common vertex but they have one arm
common.
‘b and ‘c are also a pair of alternate angles.
A pair of alternate angles made by a transversal with parallel lines are
always equal.
?‘a = ‘d and ‘b = ‘c.

‘a and ‘b are alternate angles. So, ‘a = ‘b.

(iii) Corresponding angles
‘a and ‘b are a pair of corresponding angles. One of them is exterior and
other is interior. They are lying to the same side of the transversal and they
do not have any common vertex and arm. ‘c and ‘d are also a pair of
corresponding angles.

A pair of corresponding angles made by a transversal with parallel lines
are always equal.

? ‘a = ‘b and ‘c = ‘d.

‘a and ‘b are corresponding angles. So, ‘a = ‘b.

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 199 Vedanta Excel in Mathematics - Book 6

Geometry – Angles

Worked-out examples

Example 1: If xq and 50q are co–interior angles, find the size of xq.
Solution:
Here, xq + 50q = 180q [The sum of a pair of co–interior angle is 180q]
or, xq = 180q – 50q = 130q

Example 2: In the given figure, find the size of ‘BDC.

Solution:

Here, ‘BAC + ‘ABD = 180q [The sum of
co– interior angle is 180q]

or, 70q + ‘ABD = 180q
or, ‘ABD = 180q – 70q = 110q
Again,‘ABD + ‘BDC = 180q [The sum of co–interior angle is 180q]
or, 110q + ‘BDC = 180q
or, ‘BDC = 180q – 110q = 70q

Example 3: In the figure alongside, show that
(a) ‘a = ‘g (b) ‘e = ‘y.

Solution:

a) i) ‘a = ‘c [vertically opposite angles]
ii) ‘c = ‘g [corresponding angles]
iii) ? ‘a = ‘g [From (i) and (ii)] proved.
[corresponding angles]
b) i) ‘e = ‘p [alternate angles]
ii) ‘p = ‘y [from (i) and (ii)] proved.
iii) ? ‘e = ‘y

EXERCISE 13.3
General Section – Classwork
1. Let's look at the adjoining figure, then tell and write the answers in the

blanks as quickly as possible.

a) ‘b and .............. are corresponding angles.
b) ‘c and .............. are alternate angles.
c) ‘e and .............. are co- interior angles.
d) ‘h and ............. are alternate exterior angles.
e) ‘a and ‘e are .................................. angles.
f) ‘d and ‘f are ................................ angles
g) ‘c and ‘f are .................................. angles.

Vedanta Excel in Mathematics - Book 6 200 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur