Vedanta Excel in Mathematics Book 6 mixed Units

Algebra

A power is a product of repeated multiplication of the same base. The exponent
of a power is the number of times the base is multiplied.

5a3 x×x×x
Exponent
Base x3 exponent
Coefficient base

power

Let’s take another algebraic term y.

Here, y = 1.y1. So, in y, its coefficient is 1 and exponent is also 1.

If the coefficient of a base is a number, it is called a numerical coefficient. In 3x,
3 is the numerical coefficient of x. If the coefficient of a base is a letter, it is called
a literal coefficient. In ax, a is the literal coefficient of x.

EXERCISE 11.1

General Section – Classwork

1. Let’s tell and write whether the letters represent ‘constant’ or ‘variable’.

a) x represents the natural numbers between 3 and 5. x is a ........................

b) x represents the natural numbers between 6 and 9. x is a ........................

c) y represents the even prime numbers. y is a ........................

d) y represents the composite numbers less than 10. y is a ........................

2. Let’s tell and write whether the following expressions are monomial,
binomial or trinomial.

a) 7xyz is ......................................... expression.

b) 2a + 5b – c is a ......................................... expression.

c) xy – ab is a ......................................... expression.
pq ......................................... expression.
d) r is a ......................................... expression.

e) 5x ÷ y is a

3. a) In 5a2, coefficient is .............. base is .............. power is ..............
b) In 4x3, coefficient is .............. base is .............. power is ..............
c) In y, coefficient is .............. base is .............. power is ..............
d) In 6ax, numerical coefficient is ............... literal coefficient is ..............

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

a) Sum of a and b = .........................................

b) Difference of a and b = .........................................

c) product of a and b = .........................................

d) The quotient of a divided by b = .........................................

e) Two times the sum of x and y = .........................................

f) Five tomes the product of m and n = .........................................
Creative Section

5. Let’s take the terms x, y and z. Make monomial, binomial and trinomial
expressions of your own using these terms.

6. Rewrite the following statements in algebraic expressions.

a) Product of x and y is added to z.
b) Three times the sum of x and y is increased by 5.
c) Two times the difference of p and q is decreased by 4.
d) Product of p and q is subtracted from r.
e) Five times the product of a and b is increased by x
f) The sum of x and y is divided by 2 and decreased by 7.

7. a) The present age of Anamol is x years.

(i) How old was he 2 years before?
(ii) How old will he be after 2 years ?
(iii) If his father is four times older than him, how old is his father ?

b) The breadth of a room is b metre. If its length is 5 metres longer than its
breadth, represent the length of the room by an expression.

c) The marks obtained by A in maths is x. The marks obtained by B is double
than that of A and marks obtained by C is double than that of B. Represent
the marks obtained by B and C by expressions.

8. Rewrite the following formulae in algebraic expressions.

a) The perimeter of a triangle is sum of its three sides a, b and c of the
triangle. What is the formula of perimeter of the triangle ?

b) The area of a triangle is half of the product of base (b) and height (h).
What is the formula of area of the triangle ?

c) The perimeter of a square is four times of its length (l). What is the
perimeter of the square?

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d) The area of a square is the square of its length (l). What is the area of
the square?

e) The perimeter of a rectangle is two times the sum of its length (l) and
breadth (b). What is the perimeter of the rectangle ?

f) The area of a rectangle is the product of its length (l) and breadth (b).
What is the area of the rectangle ?

11.4 Evaluation of algebraic expressions
We obtain the value of a term or an expression by replacing the variable/s
of the term or expression with numbers. It is called evaluation of a term or
expression. For example:

If x = 2 and y = 3, then 4x = 4 × 2 = 8, 2(x + y) = 2(2 + 3) = 2 × 5 = 10

(xy)2 = (2 × 3)2 = 62 = 36, (x – y)2 = (2 – 3)2 = (– 1)2 = 1 and so on.

Worked-out examples

Example 1: If y = 2x + 1 and x is a variable of the set B = {1, 2, 3}, find the
possible values of y.

Solution :
When x = 1, then y = 2x + 1 = 2 × 1 + 1 = 2 + 1 = 3
When x = 2, then y = 2x + 1 = 2 × 2 + 1 = 4 + 1 = 5
When x = 3, then y = 2x + 1 = 2 × 3 + 1 = 6 + 1 = 7

So, the required values of y are 3, 5 and 7.

Example 2: If a = 3b, express 4a + 5b in terms of b and evaluate the
expression when b = 2.

Solution :

Here, a = 3b

∴ 4a + 5b = 4 × 3b + 5b = 12b + 5b = 17b

Now, when b = 2, then 17b = 17 × 2 = 34

Example 3: If x = 2y = 4z, express x + 3y + z in terms of z and evaluate the
expression when z = 3.

Solution :

Here, x = 2y = 4z

∴ x = 4z and 2y = 4z
4z
y = 2 = 2z

Now, x + 3y + z = 4z + 3 × 2z + z = 11z

Again, when z = 3, then 11z = 11 × 3 = 33

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EXERCISE 11.2
General Section – Classwork
1. Let's input x = 1, 2, 3, ... Then tell and write outputs.

Input (x) Outputs
1
X+1 X+5 2X 3X
................. .................
................. ................. .................
.................
2 ................. ................. ................. .................
.................
3 ................. ................. .................

4 ................. ................. .................

5 ................. ................. .................

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

a) If x = 2 and y = 3, then 4x = ..........., 5y = ........... , 2xy = ...............

b) If x = – 1 and y = 2, then x + y = ................, x – y = ...............

c) If a = 3 and b = 2, then (ab)2 = .............., (a + b)2 = ...............

d) If l = 5 and b = 4, then l × b = ................, 2 (l + b) = ...............

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

a) If x = 2y, then x + y in terms of y = .......................

b) If x = 3y, then 2xy in terms of y = ......................

c) If a = 2b and b = 3, then a + b = ................

d) If p = 3q and q = 2, then p – q = ....................

Creative Section - A
4. If x = 2 and y = 3 and z = 4, evaluate the following expressions.

a) x + y + z b) 2x + 3y – z c) 3(x – y + z) d) 5xy
e) x2 + y2 f) y2 + z2 – x2
g) x+ 2y h) (x–y+z)2
z y

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5. If = 3, = 4 and = 5, find the values of following expressions.

a) 2 – b) 3 + 2 – c) ( + ) d) ( + ) ÷

6. a) The area of a square = l2. Find the area of squares in sq. cm.

(i) l = 4cm (ii) l = 7cm (iii) l = 3.5 cm (iv) l = 4.2 cm

b) The perimeter of a square = 4l. Find the erimeter of squares in cm.

(i) l = 6cm (ii) l = 9cm (iii) l = 8.5 cm (iv) l = 4.3 cm

7. a) Area of rectangle = l × b. Find the area of rectangles in sq. cm.
(i) l = 7cm, b = 4cm (ii) l = 8cm, b = 6cm (iii) l = 7.5 cm, b = 4cm

b) perimeter of rectangle = 2 (l + b). Find the perimeter of rectangles in cm.
(i) l = 6cm, b = 4cm (ii) l = 9cm, b = 5cm (iii) l = 6.4cm, b = 3.7 cm

8. The area of four walls of a room = 2h(l + b). Find the area of four walls in sq. m.

(i) l = 10 m, b = 8 m, h = 5 m (ii) l = 12 m, b = 7.5 m, h = 4 m

9. a) The area of a circle = Sr2, where S = 22 . Find the area of circles in sq. cm.
7

(i) r = 7 cm (ii) r = 14 cm (iii) r = 21 cm

b) The perimeter of a circle = 2Sr, where S = 22 . Find the perimeter of circle
in cm. 7

(i) r = 7 cm (ii) r = 14 cm (iii) r = 21 cm

10. a) The volume of a cube = l3. Find the volume of cubes in cubic cm.

(i) l = 2 cm (ii) l = 4 cm (iii) l = 5 cm

b) The area of a cube = 6l2. Find the area of cubes in sq. cm.

(i) l = 3 cm (ii) l = 5 cm (iii) l = 6 cm

11. a) The volume of a cuboid = l u b u h. Find the volume of cuboids in cubic cm.

(i) l = 5 cm, b = 4 cm, h = 3 cm

(ii) l = 10 cm, b = 7 cm, h = 5.5 cm

b) The area of a cuboid = 2(lb + bh + lh). Find the area of cuboids in sq. cm.

(i) l = 8 cm, b = 5 cm, h = 2 cm

(ii) l = 10 cm, b = 6 cm, h = 4 cm
Creative Section - B
12. If x = 5 cm, find the length of the following line segments.

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13. Write the algebraic expressions to represent the perimeters of the following
figures. If x = 2, y = 3 and z = 5, find the perimeters of the figures.

14. a) x is a variable on the set A = {1, 2, 3}, that is, x can be replaced by 1, 2

and 3. Evaluate the expressions (i) x + 5 (ii) 2x – 1

b) If y = 2x + 1 and x is a variable on the set B = {2, 4, 6}, find the possible

values of y.

15. a) If x = 2y, express 2x + 5y in terms of y and evaluate the expression when
y = 3.

b) If a = b + 3, express a + b in terms of b and evaluate the expression when
b = 3.

c) If x = 2a + 1, show that 3x – 6a + 7 = 10

11.5 Like and unlike terms

1 apple and 2 apples are like (same) type of things. Similarly, x and 2x are like terms.

3 pens and 4 pens are like (same) type of things. Similarly, 3y and 4y are like terms.

The algebraic terms having the same base and equal power are called like terms.
For examples:

3x, 5x, 8x, etc. are like terms because they have same base x and equal power 1.

5a2, 9a2, 12a2, etc. are like terms because they have the same base a and equal
power 2.

2 apples and 2 pens are unlike (different) types of things. Similarly, 2x and 2y are
unlike terms. 2x and 2y have unlike bases.

Again, 2x, 3x2, 5x2 etc. have the same base x but each base does not have equal
power. So, they are also unlike terms. Similarly, 5x2, 3y2, 4z2, etc. are unlike
terms because these terms do not have the same base, even though each base
has the equal power.

Thus, the algebraic terms which have different bases or different powers are
called unlike terms.

11.6 Addition and subtraction of algebraic terms

Let's study the following illustrations and learn to add coefficients while adding
like terms.

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2 apples + 3 apples = 5 apples 2x + 3x = 5x

4 oranges + 5 oranges = 9 oranges 4y + 5y = 9y

6 pens – 2 pens = 4 pens 6a – 2a = 4a

8 books – 3 books = 5 books 8p – 3p = 5p

Thus, to add or subtract algebraic terms, we should add or subtract the coeffi-
cients of like terms. For example:

2x + 3x = 5x, 4x2 + 6x2 = 10x2, 3ab + 2ab + ab = 6ab
8y – 5y = 3y, 9y3 – 4y3 = 5y3, 7pqr – 6pqr = pqr

But, we cannot add or subtract the coefficients of unlike terms.

For example:

2x + 3y + x + 4y = 3x + 7y, 4a + 5a + 8 = 9a + 8 and so on.

EXERCISE 11.3
General Section A – Classwork
1. Let’s tell and write whether these terms are like or unlike.

a) 4x, 7x, – 3x ................. terms b) 3x, 2x2, xy ................. terms

c) 3a2, 4a3, 6a ................. terms d) p2, 3p2, 8p2 ................. terms

e) 2xy, 5xy, 3xy ................. terms f) 2x2y, 5xy2, 3 2y2 ................. terms

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

a) a + 2a = ........... b) x2 + x2 = ............ c) 3xy + 2xy = ...........

d) 5x – 3x = ........... e) 4a2 – a2 = ............ f) 9pq2 – 4pq2 = ..........

3. a) What should be added to 2x to get 5x? ..................

b) What should be subtracted from 8x to get 3x ? ..................
Creative Section

4. Add or subtract.

a) 4x + 7x b) 5a2 + 9a2 c) 5xy + 3xy d) 3abc + 6abc

e) 10p3 – 3p3 f) 12mn – 7mn g) 9x2y – 4x2y h) 8xy2 – 7xy2
5. Simplify

a) a + 2a + 3a b) 2p2 + 3p2 + p2 + 4p2

c) 2x + 3y + x + y d) 6m + 2n + n – 4m
e) 3xy + yz – xy – 2yz – 5 f) 5p2 – 2q2 + 2p2 – 3q2

6. a) What should be added to 9x to get 15x ?

b) What should be added to p to get p + q ?
c) What should be subtracted from 12a to get 5a ?
d) What should be subtracted from 3x to get 3x – 2y ?

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7. Game Time
Complete this obstacle course.

START x Add 4x Subtract 6x Add 5x
Subtract x
Subtract 3x Same answer in
Add 7x both ways? If
not, start again.

Add 6x

Subtract 2x Add 9x Subtract 3x FINISH

11.7 Addition and subtraction of algebraic expressions

While adding or subtracting two or more binomials, trinomials or multinomial,
the like terms should be arranged in the same column, then, their coefficients
should be added or subtracted. Alternatively, we can arrange the expressions
horizontally and the addition or subtraction of like terms can be performed.

Worked-out examples

Example 1: Add 7x2 + 4xy + 5y2 and 2x2 – 3xy – 8y2.

Solution:

Addition by horizontal arrangement Addition by vertical arrangement

7x2 + 4xy + 5y2 + 2x2 – 3xy – 8y2 7x2 + 4xy + 5y2

= 7x2 + 2x2 + 4xy – 3xy + 5y2 – 8y2 2x2 – 3xy – 8y2

= 9x2 + xy – 3y2 9x2 + xy – 3y2

Example 2: Subtract 2ab – 4bc + 7 from 5ab – 4bc – 2.

Solution:

Subtraction by horizontal arrangement Subtraction by vertical arrangement
5ab – 4bc – 2 – (2ab – 4bc + 7) 5ab – 4bc – 2

= 5ab – 4bc – 2 – 2ab + 4bc – 7 ”2ab 4bc ” 7

= 5ab – 2ab – 2 – 7 = 3ab – 9 3ab – 9

Example 3: What should be added to 3x – 4 to get 8x + 5?

Solution: Let’s think, what should be
added to 7 to get 10? It’s 3 and
The required express to be added is it is 10 – 7.
8x + 5 – (3x – 4) It’s my good idea to work out
such problems!
= 8x + 5 – 3x + 4
= 8x – 3x + 5 + 4 = 5x + 9

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Example 4: What should be subtracted from 8x – 3y + 2 to get 5x + 2y – 1?

Solution:

The required expression to be subtracted is

8x – 3y + 2 – (5x + 2y – 1) Let’s think, what should be
= 8x – 3y + 2 – 5x – 2y + 1 subtracted from 9 to get 7? It’s 2
= 8x – 5x – 3y – 2y + 2 + 1 and it is 9 – 7.
= 3x – 5y + 3
I can also work out such questions.

Example 5 : If x = m + 2 and y = n – m, show that x + y = n + 2.

Solution :

Here, x = m + 2 I got it !

y=n–m In x + y, x is replaced by m + 2

Now, x + y = m + 2 + n – m and y is replaced by n – m.

= n + 2 proved.

EXERCISE 11.4
General Section – Classwork
Let’s tell and write the correct answers as quickly as possible.

1. a) Sum of x + 1 and x + 2 = ............................................

b) Sum of 3x + 5 and x – 4 = ............................................

c) Sum of a2 + b2 and a2 – b2 = ............................................

d) Difference of 2x + 3 and x + 1 = ............................................

e) Difference of 4y + 7 and 3y + 2 = ............................................

f) Difference of 8a + 5 and 4a – 3 = ............................................

2. Complete the following table.

+ a+2 a–1 a+b

a+5 2a + b + 5
a–3
a–b 2a – 4

2a – b + 2

Creative Section - A b) 2x + 5 and 3x + 1 c) 3a + 8 and 4a – 5
3. Add. e) 9m – 2 and m – 7 f) 11 – 2n and 4 – 5n
h) 8a – 5b and a – 3b i) p2 + q2 and 2p2 + 3q2
a) x + 3 and x + 2
d) 4 + 3a and 5 – 8a
g) 3a + 4b and 7a + 2b

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j) 3a2 – 5b2 and 2a2 + 7b2 k) x2 – xy + y2 and 2x2 + 6xy – y2

l) x2 + x – 2 and 2x2 – 5x + 7 m) 2ab – 3bc + 4 and ab + 4bc – 5

n) 8abc – 5ab + 3a and 2a + 7ab – 4abc

4. Subtract.

a) x + 1 from 5x + 4 b) 2x + 3 from 5x + 1

c) 6a – 7 from 9a + 2 d) 2a + 5 from 6a – 1

e) 3p – 2 from 5p – 7 f) 8 – 3p from 1 – p

g) 5m + 3n from 7m – 2n h) 9mn – 5p from 6mn – p

i) 4x2 – 3y2 from 9x2 + 4y2 j) 3a3 + 5b3 from 8a3 – b3

k) 2x3 – 5x2 + x – 6 from 4x3 + 2x2 – 3x + 7

l) 4ax – 6by + 7 from 6ax + 2by – 3

m) abc + 2ab – 3bc – ca from 3ab – ab + bc + 2ca

n) xyz – 4xy + 5 from 5xyz + xy + 3yz – 4

5. a) What should be added to 2x – 3 to get 5x + 7?
b) What should be added to 3a + 4b to get 7a – 5b?
c) What should be subtracted from 5m – 3n to get 2m + 5n?
d) What should be subtracted from 5x – 3y to get 2x – 5y?
e) To what expression 2a – 3b + 1 must be added to get 4a + 7b – 3?
f) From what expression x2 + 5y2 – 3xy must be subtracted to get 2x2 – y2 + 4xy?

Creative Section - B

6. a) If x = a + 7 and y = b – a, show that x + y = b + 7.

b) If x = 2m – n and y = m + n, show that x – y = m – 2n.

c) If x2 = 2a2 – b2, and y2 = 2b2 – c2 and z2 = 2c2 – a2,

show that: x2 + y2 + z2 = a2 + b2 + c2.

7. Simplify:

a) 4x + 3y – (3x + y) b) 5a – 3b – (a + 6b)
c) 8p + q – (5p – 3q) d) 6a – 5x – (2x – 3a)
e) – 7m – 2n – 5 – (–8m – n +1) f) –a – 2t – (t – 4a) – (–3a – 5t)
g) –(b – z) – (–2b + z) – (–3b – z) h) –(a – b) – (–4b + 3c) – (c – 2a)

8. a) The sides of a triangle are x + 4, 2x – 3 and 3x + 1, find its perimeter.
b) The sides of a triangle are 2x + 3y, x + 2y and 7x – 2y, find its perimeter.

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11.8 Multiplication of algebraic terms

While multiplying the algebraic terms, the coefficients of the terms are multiplied
and the exponents of the same bases are added. For example:

Example 1: Multiply 3x by 2x. 3 × 2 = 6 (Coefficient are
Solution: multiplied.)
Here, 3x u 2x = 6x2 x × x = x1+1 = x2 (Exponents of
the same bases are added.)
The multiplication of 3x × 2x can also be
shown diagrammatically. x2 x2 x2 x
From diagram, 2x
3x × 2x = x2 + x2 + x2 + x2 + x2 + x2 = 6x2
x2 x2 x2 x
Example 2: Multiply 4a2b by – 5ab2 x xx
Solution:
Here, 4a2b u (– 5ab2) = – 20a3b3 3x

I got it!
4 × (–5) = –20
a2 × a = a2 + 1 = a3
b × b2 = b1 + 2 = b3

11.9 Multiplication of polynomials by monomials

Of course, monomial, binomial and trinomial with positive exponents of variables
are also called polynomials. To multiply a polynomial by a monomial, each term
of the polynomial is separately multiplied by the monomial. For example:

Example 3: Multiply: (a + b) × c. ac + bc c
Solution:
ac bc
(a + b) × c = a × c + b × c ab
= ac + bc a+b

Example 4: Multiply 4a2 + 3b2 by 2ab

Solution: It’s easier!
Each term of 4a2 + 3b2 is
Here, 2ab × (4a2 + 3b2) = 2ab × 4a2 + 2ab × 3b2 separately multiplied by 2ab

= 8a3b + 6ab3.
Example 5: Multiply 7x2y2 – 4xy + 3 by – 2xy

Solution:

Here, –2xy(7x2y2 – 4xy + 3) = –2xy × 7x2y2 – 2xy(–4xy) – 2xy × 3
= –14x3y3 + 8x2y2 – 6xy

Example 6: If x = p + 3 and y = 2p, show that xy = 2p2 + 6p.

Solution: It’s easier!
Here, x = p + 3 and y = 2p In xy, x is replaced by p + 3 and

∴ xy = (p + 3) × 2p y is replaced by 2p.

= 2p × p + 2p × 3 = 2p2 + 6p proved.

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Example 7: If x = 2a, y = 3a and a = 4, find the value of 6xy .
Solution:
Here, x = 2a and y = 3a

∴ 6xy = 6 × 2a × 3a = 36a2 = 6a
When a = 4, then 6a = 6 × 4 = 24

EXERCISE 11.5
General Section – Classwork

1. Area of rectangle = length (l) × breadth (b). Let’s find the area of the
following rectangles.

a) b)

3y Area (A) = ............ 5q Area (A) = ............

4x

c) d) 2p (2x+1) 2a Area (A) = ............
2x Area (A) = ............ 5a Area (A) = ............
3x
3x

e) f)
x Area (A) = ............

(2x + 1)

2. Volume of cuboid (V) = length (l) × breadth (b) × height (h). Let’s find
the volume of following cuboids.

a) b) 3z

c

b 2y
a 4x

V = ..................... V = .....................
c) d)

x 2p

2x 3p
2x 5p

V = ..................... V = .....................

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3. Let's tell and write the products as quickly as possible.

a) x × x = ........... b) 2x × x = .......... c) 3a × 2a = ...........

d) p2 × p = .......... e) 3b2 × 4b2 = ........... f) 2x3 × 5x2 = ...........

g) x (x + 1) = ................... h) 2x(x – 3) = ...................

i) 3x(x2 + 2) = ................... j) 2a(a2 – 5) = ...................

4. Look at the targets and investigate the operation. Tell and write the

answers as quickly as possible. y a
y
a y
x xa a

2x 4a aa
x2 a4

a

x 2p y 2x
2y 3y
2x

xx p 3x
xx 3x x

Creative Section - A

5. Multiply.

a) x × x b) 2x × 3x c) 3x × 4x

d) x × 2x2 e) 2y × 3y2 f) 3xy × 2xy

g) 4a2b × 3ab2 h) –5xyz × 4xyz i) 3xy × (–5x2y2)

6. Simplify.

a) x u x u x2 b) 2a2 u 3a u 5a c) 3x u 2y u 2x u y

d) p2 u 5q u 2p u 3q2 e) 3b3 u c2 u 2b2 u 4c3 f) xyz u 2xy u yz u 5zx

g) (– 3ab) u (– 2bc) u (– abc) h) (–5qr) × (– 2pqr) u pq

i) 6x2y u (– xy2) u xyz u (– 3yz2)

7. Simplify.

a) 2x × 2x + 2x × 5 b) 3a × 2a – 3a × 4 c) 5x2 × x2 + 5x2 × 8

d) ab × 2a – ab × 3b e) ab × 5a2 – ab × 3b2 f) 10x2y2 × 3x – 10x2y2 × 2y

g) –2x × 4x – (–2x) × 3 h) –pq × 2p – (–pq) × 3q i) –3abc × 2a2 + (–3abc) × 5bc

8. Multiply.

a) 3x (2x + 5) b) 4a (5a – 6) c) 6p (8p2 – 1)

d) 2y2 (y2 + 7) e) 6m2 (2 – 3m2) f) 7b3 (8 + 3b2)

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g) xy (x – y) h) 2xy (2x + 3y) i) 2ab (3a2 – 4b2)
j) – 3pq (2p + 5q) k) – 2bc (7b – 3c) l) – 5xy (5x2 – 4y2)
m) – 4x (x2 – 2x + 1) n) – 5a (3a2 + 3a – 5) o) – 3xy (4x2 – 5xy – 3y2)

Creative Section - B
9. a) If x = a + 5 and y = 2a, show that xy = 2a2 + 10a.

b) If p = 5x and q = 3x – 1, show that 2pq = 30x2 – 10x.
c) If a = x – 3, b = 3x and x = 2, find the value of 4ab.
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)

11.10 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

6m2 + 5mn – 6n2

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Example 2: Multiply (2a – 3b + 5) by (4a – b).

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

EXERCISE 11.6

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

i) p (7p – 2q) + 4q (7p – 2q)

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

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.

11.11 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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11.12 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 –

12.13 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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EXERCISE 11.7
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) ....... ÷ ....... = .......

Vedanta Excel in Mathematics - Book 6 164

Unit Geometry: Point and Line

14

14.1 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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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.

14.2 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

14.3 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.

14.4 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 14.1

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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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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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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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

15

15.1 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.

15.2 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

15.3 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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15.4 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 15.1
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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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°

15.5 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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(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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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 15.2
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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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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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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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 Quiz time!
13. Tick (¹) 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) 45° (iii) 60° (iv) 90°

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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.

15.7 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.

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(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.

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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 15.3
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.

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2. a) If x° and y° are co-interior angles between parallel lines,
then x° + y° = .................

b) If x° and 80° are co - interior angles between parallel lines,
then x° = ......................

c) If y° and 45° are alternate angles between parallel lines, then y° = ..........
d) If z° and 120° are corresponding angles between parallel lines,

then z° = ....................

Creative Section
3. Find the sizes of unknown angles.

4. Find the sizes of unknown angles.

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5. From the figure given alongside, show that

i) ‘w = ‘c ii) ‘x = ‘s iii) ‘y = ‘g

iv) ‘a = ‘r v) ‘d = ‘s vi) ‘p = ‘c

6. In the adjoining figure, if ‘x + ‘a + ‘y = 180q,
show that ‘a + ‘b + ‘c = 180q.

7. a) In the given figure, ‘ABC = 50q,
‘ACE = 60q and AB//EC. Find the sizes of
‘BAC, ‘DCE and ‘ACB.

b) In the figure alongside, PQ//RS. Find the
values of xq and yq.

c) In the adjoining figure, CE//AB and AC//BF.
Find the sizes of wq, xq, yq and zq.

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