Vedanta Excel Maths Book 7

Equation, Inequality and Graph

(i) input (x) (ii) input (x) (iii) input (x)

Rule output Rule output Rule output
+4 (y) –3 (y) ×1 (y)

(iv) input (x) (v) input (x) (vi) input (x)

Rule output Rule output Rule output
×2+1 (y) ×3–1 (y) ×4+3 (y)

b) If x and y represent input and output numbers respectively, write the relation
between x and y in each of Q. No. 4 a).

input (x)

5. a) From the given function machine, write down the

mathematical relation between x and y. If the input is the Rule
set of integers between – 3 and 3, show the outputs in a ×3–2
table. Also, show the relation by an arrow-diagram.
output (y)

b) From the function machine given alongside, write down input (x)
the mathematical relation between the x and y. If the
input is the set of integers between –4 and 4, show Rule
the outputs in a table. Also, show the relation by an ×2+3
arrow-diagram.
output (y)

6. Let's copy the tables given below. Discover the relation between x and y.
Express the relation mathematically and use the relation to find the missing
outputs.

a) Input (x) 0 1 2 3 4 b) Input (x) 2 3 4 5 6
Output (y) 2 3 4 ? ? Output (y) 1 2 3 ? ?

c) Input (x) 1 2 3 4 5 d) Input (x) 0 1 2 3 4
Output (y) 2 4 6 ? ? Output (y) 1 3 5 ? ?

7. From the arrow-diagrams given below, write down the relation between x and
y mathematically. Also, express the input and the output numbers in the form
of ordered pairs and draw graphs to show the relation between x and y.

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 199 Vedanta Excel in Mathematics - Book 7

Equation, Inequality and Graph

a) x y b) x y c) x y

0 2 1 2 1 3
1 3 2 4 2 5
2 4 3 6 3 7
3 5 4 8 4 9
5 11

8. Let's copy the tables given below. Write the missing outputs by using the given
relations between x and y. Also draw graphs to represent each relation.

a) x 0 1 3 –1 –3 b) x 1 2 3 45

y=x+2 2 3 ? ? ? y=x–2 ? 0 ? ? ?

c) x –2 0 2 6 9 d) x –3 –1 0 4 7

y=x+3 ? ? 5 ? ? y = 9 – x 12 ? ? ? ?

e) x –1 –4 0 2 5 f) x –3 –2 1 3 6
y = 2x+1 ? ? ? ? 11 y = 3x – 2 ? –8 ? ? ?

9. Express the following relations between input (x) and output (y) mathematically.
Input the numbers from 0 to 4 in the relation and find outputs. Then, express
(input, output) in ordered pairs.

a) Output is 1 more than two times the input.

b) Output is 1 less than two times the input.

c) Output is 2 more than three times the input.

d) Output is 2 less than three times the input.
It's your time - Project work!

10. a) Let's write any three relation between input (x) and output (y) of your own
choice. Input the numbers from –3 to +3 in each relation and obtain outputs.

Show the inputs and outputs in

(i) ordered pairs

(ii) tables

(iii) arrow-diagram

b) If the rate of cost of any item is the input (x), total cost is the output (y) and
the number of items in 'n', write the relation between x and y.

Vedanta Excel in Mathematics - Book 7 200 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Unit Coordinates

12

12.1 Coordinates – Looking back
Classroom - Exercise

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

a) The x and y-coordinates of the point (2, – 5) are ........... and ...........

b) If x-coordinate of a point is 3 and y-coordinate is – 2, the coordinates of the
point are ...................

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

d) The y-coordinate of a point on x-axis is ...................

e) The x-coordinate of a point on y-axis is ...................

f) The point (0, 5) lies on ................... axis

g) The point (4,0) lies on ................... axis.

h) The point (1,2) lies in the ......................................... quadrant.

i) The point (5, 3) lies in the ......................................... quadrant.

j) The point ( 3, 4) lies in the ......................................... quadrant.

k) The point ( 7,4) lies in the ......................................... quadrant.

2. From the given graph let's tell and write the coordinates of the points as quickly
as possible. Y

Coordinates of A are ........................... B A
Coordinates of B are ............................. OX
Coordinates of C are ............................... X'
Coordinates of D are ............................... C D

201Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Y'
Vedanta Excel in Mathematics - Book 7

Coordinates

In the given graph, point P is at a distance of 5 units along Y
to the direction of OX and 3 units up to the direction of OY
from the point O.

Here, the position of the point P on the graph is 5 units along 4
and 4 units up which is written as (5, 4).

Thus, the ordered pair of numbers which are used to indicate O 5 X
the position of a point on a grid is known as the coordinates
of the point.

Here, 5 is called x-coordinate or abscissa.

4 is called y-coordinate or ordinate.

The system of describing position of a point by using coordinates was invented by a
French Mathematician called René Descartes in the 17th century.

12.2 Coordinate axes and quadrants Y

In the figure alongside, XOX’ and YOY’ are 2nd 1st
horizontal and vertical number lines respectively. Quadrant Quadrant
They are intersecting each other perpendicularly
at the point O. These number lines are called the X' O X
coordinate axes. 3rd
Quadrant 4th
Here, XOX’ is called the x-axis Quadrant

YOY’ is called the y-axis

The point O is called the origin.

Furthermore, the intersection of x-axis and y-axis Y'

divide the coordinate plane into 4 regions. These regions are called the quadrants.

The region XOY is called the first quadrant.
The region X’OY is called the second quadrant.
The region X’OY’ is called the third quadrant.
The region XOY’ is called the fourth quadrant.

12.3 Finding points in all four quadrants

The co-ordinates of a point is always written as Y
the ordered pair of (x-coordinate, y-coordinate),
i.e. (x, y). Now, let’s learn to find out the B (-2, 4)
coordinates of points in different quadrants.
A (4, 2)

(i) In the first quadrant, the coordinates of a X' 2nd Quadrant 1st Quadrant X
point is (x, y). (-x, y) (x, y)

For example, the coordinates of A is (4, 2) 3rd QuadrantO 4th Quadrant
(-x, -y) (x, -y)
C (-1, -2)
D (4, -3)
(ii) In the second quadrant, the coordinates of a
point is (– x, y). Y'

For example, the coordinates of B is (– 2, 4)

Vedanta Excel in Mathematics - Book 7 202 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Coordinates

(iii) In the third quadrant, the coordinates of a point is (– x, – y).
For example, the coordinates of C is (– 1, – 2)

(iv) In the fourth quadrant, the coordinates of point is (x, – y).
For example, the coordinates of D is (4, – 3)

(v) The coordinates of the origin is (0, 0).

(vi) The coordinates of any point on the x–axis is (x, 0) or (– x, 0).

(vii) The coordinates of any point on the y-axis is (0, y) or (0, – y).

12.4 Plotting points in all four quadrants

Graph paper (or square grid) is used to plot the given point in different quadrants.
On the graph paper, we should first draw the coordinate axes XOX’ and YOY’.

Study the following illustrations and learn to plot the points in different quadrants.

plotting (3, 2) plotting (– 3, 4) plotting (– 3, – 2) plotting (2, – 4)

Y Y Y Y

Q(-3,4)

P(3,2) 4

X 3 2 -3 X' X -2 -3 O X' X O2 X'

O X' X O R(-3,-2) -4

S(2,-4)

Y' Y' Y' Y'

(3, 2) lies in (– 3, 4) lies in (– 3, – 2) lies in (2, – 4) lies in
the 3rd quadrant. the 4th quadrant.
the 1st quadrant. the 2nd quadrant.

EXERCISE 12.1
General Section -Classwork
1. Let's tell and write the answers in the blank spaces.

a) A point is 4 units right from the origin along x–axis and 3 units up along
y-axis, then its coordinates are …………………

b) A point is 5 units left from the origin along x–axis and 2 units down along
y-axis, then its coordinates are …………………

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

d) If the abscissa of a point is –6 and its ordinate is 4, then its coordinates
are …………

e) The coordinates of a point (5, – 2), lies in the ………………… quadrant.

f) The coordinates of a point (– 3, – 8), lies in the ………………… quadrant.

203Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 7

Coordinates

g) The coordinates of a point are (4, 0), lies on ………………… axis.
h) The coordinates of a point (0, – 7), lies on ………………… axis.
2. Let's tell and write the coordinates of the points from the graph as quickly as
possible.

Y Coordinates of P are ....................................

Coordinates of Q are ....................................

X' O X Coordinates of R are ....................................
Coordinates of S are ....................................

Y' Y
O
3. Plot these points in the graph as quickly as
possible.

A (1, 3) , B (–3, 4)

C (–2, –4), D (5, – 2) X' X
E (2, 0) , F ( 5, 0)

G (0, 5) , H (0, 3)

Creative Section - A

4. a) Define coordinates of a point. Y'

b) What are quadrants? Express the ordered pairs of x and y-coordinates in
four different quadrants.

c) What are abscissa and ordinate? If the ordinate of a point is m and it's
abscissa in n, write the coordinates of the point.

d) What is origin? What are the coordinates of the origin?

5. Write down the coordinates of the points of intersection of each pair of straight

line segments. Y
5
a) Y b) Y c)
5 5 4

44
33 3
22 2
1
1 1 1 2 3 4 5 X X' –5 –4 –3 –2 –1–O1 1 2 3 4 5X
X' –5 –4 –3 –2 –1–O1 1 2 3 4 5 X X' –5 –4 –3 –2 –1–O1
–2
–2 –2
–3
–3 –3
–4
–4 –4
–5
–5 –5 Y'
Y' Y'

Vedanta Excel in Mathematics - Book 7 204 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Coordinates

d) Y e) Y f) Y
5 5 5

44 4

33 3
22 2
1
1 1 1 2 3 4 5 X X' –5 –4 –3 –2 –1–O1 1 2 3 4 5X
X' –5 –4 –3 –2 –1–O1 1 2 3 4 5 X X' –5 –4 –3 –2 –1–O1
–2 –2 –2
–3 –3 –3
–4 –4 –4
–5
–5 –5 Y'
Y' Y'

6. Let's plot the following points in graph papers.

a) (2, 4), (– 2, 4), (– 2, – 4), (2, – 4) b) (5, 3), (– 5, 3), (– 5, – 3), (5, – 3)

c) (6, 1), (– 6, 1), (– 6, – 1), (6, – 1) d) (7, 0), (0, 7), (– 7, 0), (0, – 7)

7. Let's plot the following points in graph papers and join them in order. Name
the geometrical shapes so formed.

a) A (2, 4), B (– 2, 3), C (0, 0) H (3, – 5)
b) P (– 5, – 2), Q (2, – 4), R (– 4, 6) D (5, 6)
c) E (2, 5), F (– 3, 4), G (– 1, 6), S (2, – 2)
d) A (4, – 2), B (– 4, – 2), C (– 2, – 4), Z (4, – 2)
e) P (2, 3), Q (– 3, 3), R (– 3, – 2), G (– 1, 3)
f) W (4, 3), X (– 4, 3), Y (–4, – 2), N (– 6, 0)
g) D (– 4, 0), E (0, 0), F (3, 3),
h) K (– 4, – 5), L (– 2, 0), M (– 4, 2),
Creative Section - B

8. Let's plot the points given in (i) and (ii) on the same axes of reference. Join them
separately by using a ruler. Find the coordinates of their points of intersection.

a) (i) (1, 0), (3, 2), (5,4) (ii) (5, 0), (3, 2), (1, 4)

b) (i) (2, – 4), (1, – 2), (– 2, 3) (ii) (4, 1), (1, – 2), (– 2, – 5)

c) (i) (1, 3), (4, 9), (– 3, – 5) (ii) (2, 2), (5, – 1), (– 2, 6)

d) (i) (2, 0), (5, 3), (– 1, – 3) (ii) (2, – 2), (3, 0), (4, 2)

9. a) P (– 3, – 1), Q (– 1, – 1), R (1, 1) are three of the four vertices of a parallelogram
PQRS and S is the opposite vertex of Q. Plot these vertices in graph paper
and find:

(i) the coordinates of the vertex S.

(ii) the coordinates of the point of intersection of the diagonals PR and QS.

205Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 7

Coordinates

b) A (– 3, – 5), B (2, – 5), C (2, 2) are three of the four vertices of a rectangle
ABCD and D is the vertex opposite to B. Plot these vertices in graph paper
and find:
(i) the coordinates of the vertex D (ii) the area of the rectangle ABCD.

c) E (2, 3), F (–3, 3), and G (–3, –2) are three of four vertices of a square EFGH
where H is the opposite vertex of F. Plot these vertices in a graph paper and
find:
(i) the coordinates of the vertex H. (ii) the area of the square EFGH.

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 mark four sets of three points such that all three points of each set lie
in the same straight line in the first, second, third, and fourth quadrants
respectively. Join these three points of each set separately to get four straight
lines in each quadrant. Write the coordinates of three points of each straight
line.

11. a) Let's draw a rectangle in a graph paper. Write the coordinates of the vertices
of the rectangle .What is the area of your rectangle?

b) 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?

12.5 Reflection of geometrical figures
Let's study the following illustration and investigate the idea of reflection.

Mirror image Object Object Mirror image Object Mirror image

In the above illustrations, the objects are reflected by the mirror and their images are

formed inside the mirror. A
In the similar way, the reflection of a B Geometrical figure
geometrical figure means formation of M
C Line of reflection
M’

the image of the figure after reflecting B' C'
the figure about the line of reflection. Mirror image

A'

Let's have a look in the following geometrical figures and learn to find the image of

corresponding objects (points, line segments, angles, figures, etc) while reflecting

on the line XX'.

Vedanta Excel in Mathematics - Book 7 206 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Coordinates

P KA

Q RL M

Q' R' L' M'

P' K' A'

In the above figure, XX' is the line of reflection. It is also called mirror line or axis
of reflection.

Here, image of point A is A', P is P', Q is Q', R is R', K is K', L is L', and M is M'.

Similarly, image of line segment, PQ is P'Q', QR is Q'R' ect. Likewise, image of 'PQR
is 'P'Q'R', image of ‘KLM is ‘K'L'M', and so on.

Again, let’s learn to draw the image of a geometrical figure reflected about a line of
reflection.

In the figure, ABC is a triangle. MM’ is the line of reflection.

(i) Draw perpendiculars AP, BQ and CR from each vertex of 'ABC on the line of

reflection. A

(ii) Produce AP, BQ and CR.

(iii) Measure the length of PA by using compasses B C

and cut off PA' = PA. Similarly, cut off QB' = QB M Q PR M’
and RC' = RC. C’

(iv) Join A', B' and C' by using a ruler. B’

Thus, 'A'B'C' is the image of 'ABC so formed. A’
Properties of reflection

(i) The geometrical figure and its image are at the equal distance from the line of
reflection.

(ii) The areas of the geometrical figure and its image are equal.
(iii)The appearance of the image of a figure is opposite to the figure.

12.6 Reflection of geometrical figures using coordinates

Here, we learn to find the coordinates of the images of geometrical figures formed
due to the reflection by x-axis and y-axis separately. In this case, x-axis and y-axis
are called the axis of reflection.

207Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 7

Coordinates

(i) x-axis as the axis of reflection

Let's study the following illustrations and learn to find the coordinates of the
image of a point in different quadrants when x-axis is the axis of reflection.

The point is in The point is in The point is in The point is in
the 1st quadrant. the 2nd quadrant. the 3rd quadrant. the 4th quadrant.

Y Y Y Y

P(2,3) P(-2,3) P(-2,-3) P(2,3)

XO X' X O X' X O X' X O X'

P'(2,-3) P'(-2,-3) P'(-2,-3) P'(2,-3)

Y' Y' Y' Y'

?P (x, y) o P' (x, –y) ? P (– x, y) o P'(–x, –y) ? P (– x, – y) o P' (–x, y) ? P (x, –y) o P' (x, y)
x-coordinate x-coordinate x-coordinate remains x-coordinate

remains the same. remains the same. the same. remains the same.
Sign of y-coordinate Sign of y-coordinate Sign of y-coordinate Sign of y-coordinate

is changed. is changed. is changed. is changed.

From the above illustrations, it is clear that, when a geometrical figure is reflected
about x-axis as the axis of reflection, the x-coordinate of the image remains the same
and the sign of y-coordinate of the image is changed.

(ii) y-axis as the axis of reflection

Study the following illustrations and learn to find the coordinates of the image
of a point in different quadrants when y-axis is the axis of reflection.

The point is in The point is in The point is in The point is in
the 1st quadrant. the 2nd quadrant. the 3rd quadrant. the 4th quadrant.

Y Y Y Y

P'(-3,2) P(3,2) P(-2,3) P'(2,3)

X' O X X' O X X' O X X' O X

Y' Y' P(-3,-2) P'(3,-2) P'(-2,-3) P(2,-3)

?P (x, y) o P' (– x, y) ? P (– x, y) o P'(x, y) Y' Y'
y-coordinate y-coordinate ? P (– x, – y) o P' (x, – y)
? P (x, – y) o P' (– x, – y)
remains the same. remains the same. y-coordinate y-coordinate
Sign of x-coordinate Sign of x-coordinate remains the same.
Sign of x-coordinate remains the same.
is changed. is changed. Sign of x-coordinate
is changed.
is changed.

From the above illustrations, it is clear that when a geometrical figure is reflected about
y-axis as the axis of reflection, the sign of x-coordinate of the image is changed but the
y-coordinate of the image remains the same.

Vedanta Excel in Mathematics - Book 7 208 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Coordinates

EXERCISE 12.2
General Section - Classwork
1. Let's observe the graph given below and name the image point of following

objects under reflection on the line MN.

B A G D Object Image
MQ C Point P
I Point A .................
Object H E Point B .................
U .................
V N Point W
P S R Image
Image
W Point E
T

Object

Line AB Line PQ ‘GHI ‘UVW

Line GH ................. ‘CED .................

Line DE ................. ‘RST .................

Line ST ................. 'CDE .................

2. Let's tell and write the coordinates of images of the given points.

a) X-axis is the axis of reflection: b)Y-axis is the axis of reflection:
P (x, y) o .......................... P (x, y) o ..........................
P (–x,y) o .......................... P (–x,y) o ..........................
P (–x, –y ) o .......................... P (–x, –y ) o ..........................
P (x, –y) o .......................... P (x, –y) o ..........................
A ( 3, 4) o .......................... A (2, 5) o ..........................
B ( 5, 2) o .......................... B ( 3, 6) o ..........................
C (–6, –1) o .......................... C (–1, –4) o ..........................
D (4, –3) o .......................... D (6, –7) o ..........................

209Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 7

Coordinates

Creative Section - A

3. a) Define transformation and write the types of transformations.

b) Write the properties of reflection.

c) After the reflection about X-axis, if the image of a point P is P'(–x, –y), in
which quadrant does P lie and what is its coordinates?

d) After the reflection about Y-axis, if the image of a point Q is Q'(x, –y), in which
quadrant does P lie and what is its coordinates?

4. If MN is the line of reflection, let's copy and draw the images of the following
figures.

a) A b) M Q N M A

c) D

B C R
M N PC

d N NB

M e) f)
P
U MN M
H

A

TS FB EN
EG

QR CD

5. Let's find the images of the following points separately under the reflection
about x – axis and y – axis.

a) (2, 4) b) (5, –3) c) (–2, 8) d) (–1, –7) e) (0, 6) f) (–9, 0 )

6. Let's copy the following figures in your own graph paper and draw their images
separately under the reflection about (i) X-axis (ii) Y-axis. Also, mention the
coordinates of the vertices of images.

a) Y b) Y c) Y

6 6 6
5 E 5 5
4 A R 4 4
F
3 Q3 3
2 2 2
1 B C P1 1
D 45 -6-5-4-3-2-1-01
X' -6-5-4-3-2-1-01 1 2 3 6 X X' 1 2 3 4 56 X X' -6K-5-4-3-2M-1---0132 1 2Y3 4 5 6 X
--32 I H Z
--23
-4 -4 -4
-5 -5 -5 X
L -6 W
-6 -6 G

Y' Y' Y'

Vedanta Excel in Mathematics - Book 7 210 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Coordinates

7. Let's plot the following points in a sheet of graph paper and draw triangles joining
the points in order. Draw the image of each triangle separately under the reflection
about (i) X-axis (ii) Y-axis and write the coordinates of the vertices of images.

a) A (2, 5), B (6, 3), C (4, 8) b) P (– 4, 6), Q (– 3, 4), R (– 1, 2)

c) X (– 6, – 4), Y (– 3, – 1), Z (– 1, – 3) d) E (2, – 7), F (5, – 3), G (2, – 1)

8. a) A (3, 6), B (2, 4), and C (5, 7) are the vertices of 'ABC. Find the coordinates of
its image under the reflection about x-axis.

b) P (– 2, 5), Q (– 4, – 3), and R (3, – 6) are the vertices of 'PQR. Find the
coordinates of its image under the reflection about x-axis.

c) X (2, 7), Y (– 4, – 8), and Z (3, – 5) are the vertices of 'XYZ. Find the
coordinates of its image under the reflection about y-axis.

d) K (– 5, 4), L (2, 6), and M (3, –3) are the vertices of 'KLM. Find the coordinates
of its image under the reflection about y-axis.

Creative Section - B

9. a) E (– 1, – 3), F (– 4, 5), and G (2, 2) are the vertices of 'EFG. Find the coordinates
b) of the vertices of 'E'F'G' under the reflection about x-axis. Again, find the
coordinates of the vertices of 'E"F"G" under the reflection about y-axis.
10. a)
b) A (2, – 7), B (– 3, – 6), and C (– 4, 2) are the vertices of 'ABC. If it is reflected
under the reflection about x-axis and then about y-axis, find the coordinates
of the vertices of 'A"B"C".

∆ ABC is reflected about the X- axis onto ∆A'B'C' with vertices A' (2, 5),
B' (5, 0), and C' (3, 1). Find the vertices of ∆ ABC.

If the points A(4, 7) and B(8, 6) are mapped on the points A' (–4, 7) and
B' (–8, 6) respectively. What is the axis of reflection?

It's your time - Project work!

11. a) Let's write large size of letters and numbers: s, v, u, 3, ª, A, B, C, D, E,

and 1, 2, 3, 4, 5 on a chart paper. Let's take the chart paper and stand in front
of a mirror facing the letters and numbers towards the mirror. Now, copy
the image of each letter and number. Then, write a short report about the
properties of reflection and present in the class.

b) Draw a triangle, a rectangle, and a square on a chart paper. Write the vertices
of each figure. Let's stand in front of a mirror facing the shapes towards
mirror. Observe how are the vertices of each figure transformed due to the
reflection of mirror. Write a short report and present in the class.

12.7 Rotation of geometrical figures

A rotation is a turn of a shape. A rotation is described by the centre of rotation, the
angle of rotation, and the direction of the rotation.

211Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 7

Coordinates

Let's study the following illustrations and investigate the idea of rotation of a point.

P' O P
90°

90°

OP P'

Point P is said to be rotated through 90q Point P is said to be rotated through 90q
in anti-clockwise direction about the in clock-wise direction about the centre
centre of rotation at O. P’ is the image of of rotation at O. P’ is the image of P so
P so formed. formed.

P' O P

180° 180°

P' O P Point P is said to be rotated through 180q
Point P is said to be rotated through 180q in clock-wise direction about the centre
in anti-clockwise direction about the of rotation at O. P’ is the image of P so
centre of rotation at O. P’ is the image of formed.
P so formed.

In the similar way, when each vertex of a geometrical figure is rotated through a
certain angle in a certain direction about a given centre of rotation, the figure is said
to be rotated through the same angle in the same direction about the same centre of
rotation.
To rotate a geometrical figure, following three conditions are required.

(i) Centre of rotation (ii) Angle of rotation (iii) Direction of rotation

A figure can be rotated in two directions.

1. Anti-clockwise direction (Positive direction)
2. Clock-wise direction (Negative direction)

In anti-clockwise direction, we rotate a figure in opposite direction of the rotation
of the second-hand of a clock. In clockwise direction, we rotate a figure in the same
direction of the rotation of the second-hand of a clock.

Now, let’s learn to draw the image of a figure when it is rotated through the given

angle in the given direction about a given centre of rotation.

Rotation through 90q in anti-clockwise direction B' Anti-clockwise

(i) Join each vertex of the figure to the centre of C' C
rotation with dotted lines.
A'
(ii) On each dotted line draw 90q at O with the help
of a protractor in anti-clockwise direction. B
O

A

Vedanta Excel in Mathematics - Book 7 212 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Coordinates

(iii)With the help of compasses, cut off OA' = OA, OB' = OB and OC' = OC.
(iv) Join A', B' and C'.

Thus, 'A'B'C' is the image of 'ABC formed due to the rotation through 90q in
anti-clockwise direction about O.

Rotation through 90q in clockwise direction O C
In this case, you should follow the similar steps as A’
mentioned above. While making the angle of 90q, you B
should draw it in clockwise direction. In the adjoining B’ Clockwise
diagram, 'A'B'C' is the image of 'ABC formed due to the A
rotation through 90q in clockwise direction about O. In C’
similar way we can rotate a geometrical figure through
180q in anti-clockwise or clockwise direction about the
given centre of rotation.

12.8 Rotation of geometrical figures using coordinates
Let's study the following illustrations and learn to find the coordinates of the image
of a point when it is rotated through 90q and 180q in anti-clockwise and clockwise
directions about the centre of rotation at origin.

(i) Rotation through 90° in anti-clockwise about origin

YY Y Y

P'(-2,3) P(-3,1) P'(1,3)
P(3,2)
O X'
XO X' X O X' X O X' X
P(3,-1)
P'(-1,-3) P(-2,-3) P'(3,-2)
Y'
Y' Y' Y'

P (3, 2) o P' (–2, 3) P (–3, 1) o P' (–1, –3) P (–2, –3) o P' (3, –2) P (3, –1) o P' (1, 3)

? P (x, y) o P' (–y, x) ? P (–x, y) o P' (–y, –x) ? P (–x, –y) o P' (y, –x) ? P (x, –y) o P' (y, x)

Thus, when a point is rotated through 90q in anti-clockwise direction about the origin
as the centre of rotation, the x and y-coordinates are exchanged by making the sign of
y-coordinate just opposite.

i.e. P (x, y) o P' (– y, x)

213Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 7

Coordinates

(ii) Rotation through 90° in clockwise about origin

Y YY Y

P(3,2) P'(1,3) P'(-3,2)
P(-3,-1)

X' O X X' O X X' O X X' O P(3,-1) X

P' (2,-3) P(-2,-3) P' (-1,-3)

Y' Y' Y' Y'

P (3, 2) o P' (2, –3) P (–3, 1) o P' (1, 3) P (–2, –3)o P'(–3, 2) P (3, –1) o P' (–1, –3)
? P (x, y) o P' (y, –x) ? P (–x, y) o P'(y, x) ? P (–x, –y) o P' (–y, x) ? P (x, –y) o P' (–y, –x)

In this way, when a point is rotated through 90q in clockwise direction about the origin
as the centre of rotation, the x and y coordinates are exchanged by making the sign of
x-coordinate just opposite.

i.e. P (x, y) o P' (y, – x)

(ii) Rotation through 180° in anti-clockwise and clockwise about origin

When a point is rotated through 180q about origin, the coordinates of the image are
the same in both directions. Study the following illustrations.

YYY Y
P'(-2,2)
P(3,2) P'(2,3)
OX
P(-3,1) P(2,-2)

X' O X X' O X X' O X X' Y'
P'(-3,-2) P'(3,-1)

P(-2,-3)

Y' Y' Y'

P (3, 2) o P" (–3, –2) P (–3, 1) o P' (3, –1) P (–2, –3) o P' (2, 3) P (2, –2) o P' (–2, 2)
? P (x, y) o P' (–x, –y) ? P (–x, y) o P' (x, –y) ? P (–x, –y) o P' (x, y) ? P (x, –y) o P' (–x, y)

Thus, when a point is rotated through 180q in anti-clockwise or in clockwise
direction about the origin as the centre of rotation, the x and y-coordinates of the
image remain same just by changing their signs.

i.e.P (x, y) o P' (– x, – y)

EXERCISE 12.3

General Section - Classwork

1. Let's tell and write the coordinates of image due to the rotation through 90° in
the given directions.

Vedanta Excel in Mathematics - Book 7 214 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Clockwise rotation through 90° Coordinates
a) P (x, y) o ....................... Anti-clock wise rotation through 90°
b) P (–x, y) o ....................... a) P (x, y) o .......................
c) P (– x, –y) o ....................... b) P (–x, y) o .......................
d) P (x, –y) o ....................... c) P (–x, –y)o .......................
e) A(–2, 3) o ....................... d) P (x, –y) o .......................
f) C (5, 2) o ....................... e) B (4, –6) o .......................
f) (–3, –1) o .......................

2. Let's tell and write the coordinates of images as quickly as possible under the
clockwise and anti-clockwise rotation through 180°.

a) P (x, y) o ....................... b) P (–x, y) o .......................

c) P (–x, –y) o ....................... d) P (x, –y) o .......................

e) A (2, 5) o ....................... f) B (–1, 4) o .......................

g) C (–3, –6) o ....................... h) D (4, –7) o .......................

Creative Section - A

3. Let's draw the images of the following figures rotating through 90q and 180q in

anti-clockwise and clockwise direction about the given centre of rotation.
G
a) C B b) P Q c)

H O

AO EF

OR

4. Let's find the images of the following points when they are rotated through 90°
in anticlockwise and clockwise direction separately about the origin as the
centre of rotation.

a) (6, 2) b) (– 3, 5) c) (1, –7) d) (–4, –9) e) (0, –5) f) (8, 0)

5. Let's find the images of the following points when they are rotated through
180° in anti-clockwise direction about origin as the centre of rotation.

a) (2, 1) b) (3, –4) c) (–5, –7) d) (–1, 6) e) (0, 6) f) (–4, 0)

6. Let's copy the following figures in your own graph paper. Draw their images
rotating through 90q in anti-clockwise direction. Also, write the coordinates of
the vertices of images.

215Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 7

Coordinates

a) Y b) 6 Y c) 6 Y
6 5 5
5 E F R
4 C B
3 4 4
3 3
2 2 2 Q
G E1
1 A 1 P
X' -6-5-4-3-2-1-01 34 X X' X X' 45 X
1 2 5 6 -6-5-4-3-2-1-01 1 2 3 4 56 -6L-5 -4 -3 -2-1-01 1 2 3 6
M N--23
--23 --32
-4 -4 -4
-5 -5 LM -5
-6 -6 K -6

Y' Y' Y'

7. Let's copy the following figures in your own graph paper. Draw their images

rotating through 180q in anti-clockwise direction. Also write the coordinates

of the vertices of images.

a) 6Y b) 6Y c) 6Y
5 5
K L 5 Y P A D E
4 B
4 4
3 3 3
22 2
M1 -6-5-4-3-2C-1-101 1
X' -6-5-4-3-2-1-01 R X X' F X X' X
1 2 3 45 6 1 2 34 5 6 -6-5 -4 -3 -2-1-01 1 2 34 5 6
S R--32 W
-2 -2
-3 -3
-4 -4 -4
-5 -5 -5 Y
T X
-6 -6 -6

Y' Y' Y'

Creative section - B

8. a) A (2, 5), B (– 3, 7), and C (2, – 4) are the vertices of 'ABC. Find the coordinates
of its image under the rotation through 90q in clockwise direction about origin.

b) X (7, 8), Y (– 5, 4), and Z (1, – 3) are vertices of 'XYZ. Find the coordinates
of 'X'Y'Z' under the rotation through 90q in anti-clockwise direction about
origin.

9. a) P (– 4, – 9), Q (1, – 6), and R (– 5, 2) are the vertices of 'PQR. Find the
coordinates of its image under the rotation through 180q in clockwise
direction about origin.

b) E (1, 5), F (– 3, 6), and G (4, – 2) are the vertices of ' EFG. Find the coordinates
of 'E'F'G' under the rotation through 180q in anticlockwise direction about
origin.

It's your time - Project work!

10. a) Let's take a rectangular sheet of cardboard paper and draw its two diagonals.
The point of intersection is the centre of the rectangle. Now, press the centre
with the tip of a pencil and rotate the rectangle:

(i) trough 90° in anticlock-wise (ii) through 90° in clock-wise

(iii) through 180° in anticlock-wise (iv) through 180° in clock-wise

b) How are the vertices, sides, and angles of the rectangle transformed due to
these rotations? Write a short report and present in the class.

c) Taking a square sheet of paper, repeat the same activities.

Vedanta Excel in Mathematics - Book 7 216 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Coordinates

11. a) Let's draw a triangle on a graph paper and write the coordinates of its
vertices. Now use the rule of rotation of each vertex through 90° in anti-
clockwise direction about origin as the centre and mark the image vertices of
the triangle. Measure the angles between the lines joining the image vertices
and corresponding original vertices. Is each angle 90°?

b) Let's repeat the similar activities for the rotation through 90° in clock-wise
direction, 180° in anticlockwise, and clockwise directions.

12.9 Displacement

The transformation of geometrical figure by moving its each point (or each vertex)

through the equal distance to the same direction is known as displacement. It is

also called translation. After the displacement, the image of any geometrical figure

so formed is congruent to the given figure. Study the following illustrations of

displacement. P A'
B'
QB B' A
C' B

A

A' C
Straight line AB is displaced to A'B'.
Here, AB = PQ = A'B'. Also, AA' // BB' // PQ 'ABC is displaced to 'A'B'C'.
Here, AA' = BB' = CC'. Also, AA' // BB' // CC'

Worked-out examples Q

Example 1: Displace the straight line PQ in the same X
P
magnitude and direction of XY. Y

Solution:

(i) From P, draw PP' // XY by using set squares. Q

(ii) From Q, draw QQ' // XY by using set squares. Q' X

(iii) Cut off PP' = XY and QQ' = XY. Y

(iv) Join P' and Q'. P

P'Q' is the required image of the line PQ. A
P' O
Example 2: Displace the triangle ABC in the magnitude and
C B
direction of OP. A
OP
Solution:

(i) From A, draw AA' // OP and cut off AA' = OP. B A' C
(ii) From B, draw BB' // OP and cut off BB' = OP. P
(iii) From C, draw CC' // OP and cut off CC' = OP.

(iv) Join A', B' and C'. B' C'

(v) 'A'B'C' is the required image of 'ABC under the displacement of the magnitude
and direction of OP.

217Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 7

Coordinates

EXERCISE 12.4

1. Let's copy and displace the following straight lines in the given magnitude and

direction. b) P c) X P
a) B

QN

PY

M Q
AQ

2. Let's copy and displace the following geometrical figures in the given magnitude

and direction. F

a) C b) X c)

P Q Q P ZD P
A B P E
R e) Y f)
d) S O X Q
C A

S

AO P R
PQ O

X B Q

It's your time - Project work!

3. a) Let's place your math book on a chart A' D'
paper and draw it's outline. Name the
vertices of the outline as A, B, C, and D. B' C'
Again, place the book on the other place of D
the chart paper and draw outline. Mark the
corresponding vertices of the outline as A', A B C
B' C', and D'. Join AA', BB', CC', and DD'
with dotted lines.

(i) Find the lengths of AA', BB', CC', and DD'. (ii) Are AA' // BB' //
CC' // DD'?

b) Repeat the similar activities tracing your geometry box, set-squares, rulers,
etc.

c) Write a short report about the observation of these activities and present in
your class.

Vedanta Excel in Mathematics - Book 7 218 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Unit Geometry: Angles

13

13.1 Angels – Looking back

Classroom - Exercise

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

a) b)
Q Name ............................ A Name ............................
B

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

O P Arms ............................. C Arms .............................

2. Let's tell and write the names and types of these angles as acute, right, obtuse,
straight, or reflex angles.
a) B b) c) Z

120° 180° 45°
OA R OP YX

‘AOB is an Obtuse angle
d) N f) O 300° G
e) A C
125° D
O T

M



In the figures given below, 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
anti-clockwise direction.

‘XOP = 90° is formed ‘XOP = 180° is formed ‘XOP = 270° is formed ‘XOP = 360° is formed

due a quarter turn due a quarter turn due a quarter turn due a quarter turn

Thus, when a fixed line segment is rotated about a fixed point, various angles are
formed at the fixed point.
Here, ‘XOP is the name of angle. O is called the vertex of angle. XO and OP are the
arms of the angle.

219Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 7

Geometry: Angles

13.2 Different pairs of angles – Review

There are various pairs of angles according to their structures and properties. For
example, adjacent angles, vertically opposite angles, complementary angles, and
supplementary angles. Let's review these various pairs of angles.

(i) Adjacent angles CC B
In the given figure, ‘AOB and ‘BOC have B A
common vertex O and a common arm OB.
They are called adjacent angles. Thus, a AO
pair of angles having a common vertex and O
a common arm are called adjacent angels.

(ii) Linear pair B

In the given figure, ‘AOB and ‘BOC are a pair of adjacent

angles. Their sum is a straight angle (180q), 180°

i.e. ‘AOB + ‘BOC = 180q CO A
‘AOB and ‘BOC are called a linear pair.

Thus, the adjacent angles in a straight line are known as a linear pair.

(iii) Vertically opposite angles

In the given figure, ‘AOC and ‘BOD are formed by C
intersected line segments and they lie to the opposite A

side of the common vertex. They are called vertically

opposite angles. ‘AOD and ‘BOC are another pair of D O B
vertically opposite angles.

Vertically opposite angles are always equal.

? ‘AOC = ‘BOD and ‘AOD = ‘BOC.

(iv) Complementary angles

In the given figure, the sum of ‘AOB and ‘BOC is a right angle C B
(90q), i.e., ‘AOB + ‘BOC = 90q A
90°
‘AOB and ‘BOC are called complementary angles. O

Thus, a pair of angles whose sum is 90q are called complementary
angles. The complementary angles may or may not be adjacent.

Here,complement of ‘AOB = 90q – ‘BOC.

complement of ‘BOC = 90q – ‘AOB

(v) Supplementary angles B

In the given figure, the sum of ‘AOB and ‘BOC is two 180°
right angles (180q),

i.e., ‘AOB + ‘BOC = 180q. CO A
‘AOB and ‘BOC are called supplementary angles.

Thus, a pair of angles whose sum is 180q are called supplementary angles. The

supplementary angles may or may not be adjacent.

Here,supplement of ‘AOB = 180q – ‘BOC

Supplement of ‘BOC = 180q – ‘AOB.

Vedanta Excel in Mathematics - Book 7 220 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Geometry: Angles

13.3 Verification of properties of angles

Activity - 1

Let's verify experimentally that the vertically opposite angles formed due to the
intersection of two line segments are equal.

Verification
(i) Let's draw three different sets of two line segments AB and CD intersecting at O.

A D D
B
O D B
C O
AO B
fig (i)
C A fig (iii) C
fig (ii)

(ii) Let's measure each pair of vertically opposite angles: ‘AOD and ‘BOC, ‘AOC
and ‘BOD with a protractor and write the measurements in the table.

Fig. Vertically opposite angles Result

‘AOD and ‘BOC ‘AOC and ‘BOD ‘AOD = ‘BOC
‘AOC = ‘BOD
(i)

(ii)

(iii)

Conclusion: From the above table, we conclude that the vertically opposite
angles formed due to the intersection of two line segments are equal.

Activity - 2
Let's verify experimentally that the sum of adjacent angles in linear pair is 180°.
Verification
(i) Let's draw three different sets of adjacent angles ‘AOC and ‘BOC in linear pair.

C C A OB

AO B AO B C
fig (i) fig (iii)
fig (ii)

(ii) Let's measure ‘AOC and ‘BOC with a protractor and write the measurements
in the table.

221Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 7

Geometry: Angles

Fig. No. ‘AOC ‘BOC ‘AOC + ‘BOC Result
(i)
(ii) ‘AOC + ‘BOC = 180°
(iii)

Conclusion: From the above table we conclude that the sum of adjacent angles
in linear pair is 180°.

Activity - 3

Let's verify experimentally that the angle formed by a revolving line in a complete
rotation at a point is 360°.

Verification
(i) Let's draw three different sets of angles ‘AOB, ‘BOC and ‘COA formed by a

revolving line OA in a complete rotation at O.

AC

B

OA OC OB

C B A fig (iii)
fig (i) fig (ii)

(ii) Let's measure ‘AOB, ‘BOC, and ‘COA with a protractor and write the

measurements in the table.

Fig. ‘AOB ‘BOC ‘COA ‘AOB + ‘BOC + ‘COA Result
No.

(i)
(ii) ‘AOB + ‘BOC + ‘COA = 360°
(iii)

Conclusion: The angle formed by a revolving line in a complete rotation at a
point O is 360°.

Worked-out examples

Example 1: If 2xq and (x + 30)q are a pair of complementary angles, find them.

Solution:

Here,2xq + (x + 30)q = 90q [The sum of a pair of complementary angles]

or, 3xq= 90q – 30q

or, xq = 60° = 20q
3

? xq = 2 u 20q = 40q and (x + 30)q = (20q + 30q = 50q

Vedanta Excel in Mathematics - Book 7 222 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Geometry: Angles

Example 2: A pair of supplementary angles are in the ratio 7 : 3, find them.

Solution:

Let the required supplementary angles be 7xq and 3xq.

Now,

7xq + 3xq = 180q [The sum of a pair of supplementary angles]

or, 10xq = 180q

or, xq = 180° = 18q
10

? 7xq= 7 u 18q = 126q and 3xq = 3 u 18q = 54q

Example 3: In the adjoining figure, find the sizes of unknown angles.

Solution: 5x°
4x°
Here, 4xq + 5xq + 6xq = 180q [Being the sum a straight angle] 6x°
or, r° p°
15xq = 180q
or, xq = 11850°= 12q q°

? 4xq = 4 u 12 = 48°, 5xq = 5 u 12q = 60q and 6xq = 6 u 12q = 72q.

Again, pq = 4xq = 48q, q = 5xq = 60q and rq = 6xq = 72q [Each pair is vertically opposite
angles]

EXERCISE 13.1
General Section - Classwork
1. Let's fill in the blanks as quickly as possible.

a) If two angles are complementary, then the sum of their measures is .................

b) Two angles are supplementary if the sum of their measures is .................

c) If a° and b° are in linear pair, then a° + b° = .................

d) The complement of 40° is .................

e) The supplement of 100° is .................

f) If p° and 80° are vertically opposite angles, then p° = .................

g) If two lines intersect at a point and one pair of vertically opposite angles are
acute angles, the other pair of vertically opposite angles are .................

Creative Section - A
2. a) Define vertex and arms of an angle with an example.

b) Define acute, obtuse, and reflex angles with examples.
c) Define adjacent angles, vertically opposite angles, complementary angles

and supplementary angles with examples.
d) Define a linear pair with an example.
e) What is the relation between a right angle and a straight angle?

223Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 7

Geometry: Angles

3. Let's state with reason whether the angles a and b are adjacent or not.
a) b) c) d)

b a ba
a b ba

4. Let's find the complements of :

a) 60° b) 80° c) 15° d) 28° e) 51°

5. Let's find the supplements of :

a) 110° b) 135° c) 150° d) 70° e) 45°

6. a) If 4x° and 5x° form a linear pair, find them.

b) If (2x – 10)° and (3x + 20)° are a linear pair, find them.

c) A pair of complementary angles are in the ratio 2 : 3, find them.

d) The difference of complementary angles is 10°. Find the measures of the
angles.

e) Find the size of an angle which is five times its supplement.

f) Find the supplementary angles in which one angle is 40° more than the
other angle

g) If the supplement of an angle is four times its complement, find the angle.

7. Let's find the sizes of unknown angles.

a) b) c) d)

110° x° 75° y° x° y°
120° 40°

e) f) g) h) (x 1)°

x° (x+30)° 3y° 2y° 2x° 3x° (2x+1)°
x° (x+20)°

i) j) k) 20° l) 3a°

(x+5)° x° z° x°
x
2x°x° y° a°
(x+25)° 4x°3x° y°
30°
m) n) 88° o) p)

p° (p+10)° y° q° x°
z° 128° 3q° (x+25)°
(p+20)° y° 140° x° 2x° 3x°
z° x° b° a° p° (x+35)°

2b°

Vedanta Excel in Mathematics - Book 7 224 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Creative Section - B Geometry: Angles

8. a) In the figure given alongside, if x = 1 y, y° x°
3
X
show that y = 135° Y

b) In the adjoining figure, ‘XOY = 2‘YOZ, and OZ

‘WOX = 3‘YOZ. Show that ‘WOX = 90q. W

c) In the figure alongside, if a + b + c = 180°, a x
show that x = a + b.
bc
d) In the given figure, if ‘a = ‘b, prove that ‘x = ‘y.
a
x
by

e) In the adjoining figure, ‘m = ‘n , show that‘p = ‘q. p
m
nq

9. a) An angle is 21° more than twice its complement. Find it.

b) An angle exceeds three times its complement by 10°, find it.

c) The supplement of an angle is 6° less than three times its complement. Find
the angle.

d) A pair of supplementary angles are in the ratio 4 : 5. Find the complement of
smaller angle.

It's your time - Project Work!

10. a) Draw the diagrams of four clocks. Then show the angles made by hour-hand
(shorter) and minute-hand (longer) at

(i) 3 o'clock (ii) 6 o'clock (iii) 9 o'clock (iv) 12 o'clock

Also, mention the angles formed due the quarter turn, half turn, three-quarter
turn, and complete turn. At what time are these angles formed?

b) Explore experimentally the relationship between vertically opposite angles
formed due to the intersection of two line segments.

c) Experimentally verify that the sum of adjacent angles in linear pair is 180°.

225Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 7

Geometry: Angles

13.4 Pairs of angles made by a transversal with parallel lines.

In the given figure, straight line PQ intersects two parallel lines A P B
AB and CD at M and N respectively. Here, the straight line PQ is C D
called a transversal . The transversal PQ makes various pairs of M
angles at M and N between the parallel lines AB and CD.
N
(i) Exterior and alternate exterior angles Q

In the given figure, ‘a, ‘b, ‘c, and ‘d are lying outside the parallel lines.

They are called exterior angles. ‘a and ‘c are lying to opposite side of the

transversal. They are called alternate exterior angles. P

Similarly ‘b and ‘d are another pair of alternate exterior A ab B
angles. M

Thus, alternate exterior angles are the pair of non-adjacent N D
exterior angles which lie to the opposite side of transversal. C dc

The alternate exterior angles made by transversal with Q

parallel lines are always equal.

? ‘a = ‘c and ‘b = ‘d

(ii) Interior and alternate interior angles P

In the given figure, ‘a, ‘b, ‘c, and ‘d are lying A M B
inside the parallel lines. They are called interior C ab D
angles. ‘a and ‘c are lying to the opposite side of the
transversal. They are called alternate interior (or simply dc
alternate) angles. Similarly, ‘b and ‘d are another pair of N
alternate angles.
Q

Thus, alternate angles are the pair of non-adjacent interior angles which lie to

the opposite sides of transversal.

Alternate angles always lie in the ' ' - shaped of parallel lines.

The alternate (interior) angles made by a transversal with parallel lines are

always equal.

? ‘a = ‘c and ‘b = ‘d P P

(iii) Corresponding angles A ab BA M B
‘a is an exterior and ‘c is an M cd

interior angles lying to the same cd DC N D
side of the transversal and they C N ab

are not adjacent to each other. Q Q

They are called corresponding

angles. ‘b and ‘d are another pair of corresponding angles.

Thus, a pair of non-adjacent interior and exterior angles lying on the same side

of transversal are said to be corresponding angles.

Corresponding angles always lie in the ' ' - -shaped of parallel lines.

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

? ‘a = ‘c and ‘b = ‘d

Vedanta Excel in Mathematics - Book 7 226 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Geometry: Angles

(iv) Co-interior angles P B
M D
In the figure given alongside, ‘a and ‘d are a pair of ab
interior angles lying to same side of the transversal. A dc
They are called co-interior (consecutive interior) angles.
Similarly, ‘b and ‘c are another pair of co-interior angles. C N
Q
Thus, a pair of interior angles lying on the same
side of transversal are said to be co-interior angles.
Co-interior angles always lie in the ' '-shaped of parallel lines.

The sum of a pair of co-interior angles made by a transversal with parallel
lines is always 180q.

Pairs of angles between parallel lines at a glance Properties
Parts of angles Diagrams

Alternate ‘a = ‘b
angles

Corresponding ‘a = ‘b
angles

Co-interior ‘a+‘b = 180°
angles

Worked-out examples

Example 1: Find the sizes of unknown angles in the following figures.

a) b)

wx 40°
100° y
z x
25°

Solution:

a) (i) w = 100q [Being vertically opposite angles]

(ii) x = w = 100q [Being corresponding angles]

(iii) y = x = 100q [Being vertically opposite angles]

y + z = 180q [Being the sum of a pair of co-interior angles]

or, 100q + z = 180q

or, z = 180q – 100q = 80q

So, w = x = y = 100q and z = 80q

227Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 7

Geometry: Angles

b) Through the point E, PQ parallel to the given parallel lines AB and CD is drawn.

(i) ‘a = 40q [Being alternate angles] A 40° B
(ii) ‘b = 25q [Being alternate angles]

‘a + ‘b + ‘x =360q [Being part of a complete turn] P ax Q
or, 40q+25q + ‘x = 360 C bE D
or, ‘x = 360q 65q 25°

? ‘x = 295q

EXERCISE 13.2
General Section - Classwork
1. From the given figure, let's tell and write the answers in the blanks spaces.

a) ‘d and ........................... are alternate angles.

b) ‘a and ............................ are corresponding angles. ab
c) ‘c and ............................ are co – interior angles. dc
d) ‘q and ‘d are ...................................... angles.
e) ‘c and ‘m are ...................................... angles. mn
f) ‘d and ‘m are ...................................... angles. qp

2. In each of the following questions, there are four options. Out of which one
option is correct. Let's tick (— ) the correct option.
a) Which of the following statement is not true?
(i) A pair of alternate angles between parallel lines are equal.
(ii) A pair of corresponding angles between parallel lines are equal.
(iii)A pair of co- interior angles between parallel lines are complementary.
(iv) A pair of co- interior angles between parallel lines are supplementary.

b) In the figure alongside, a pair of corresponding angles is ab
(i) ‘a and ‘c dc
(ii) ‘b and ‘r pq
(iii)‘d and ‘q sr
(iv) ‘p and ‘a

c) In the given figure, which of the following is true? 12
(i) ‘1 = ‘8 43
(ii) ‘4 = ‘5
(iii)‘2 = ‘7 56
(iv) ‘8 = ‘2 87

Vedanta Excel in Mathematics - Book 7 228 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Geometry: Angles

d) In the adjoining figure, which of the following is not true? ab
(i) ‘m + ‘x =180° nm
(ii) ‘a + ‘y = 180° pq
(iii) ‘n + ‘p = 180° yx
(iv) ‘a + ‘q = 180°

3. Each of the pair of these angles are formed between parallel lines. Let's tell
and write the answers as quickly as possible.

a) If x° and 45° are a pair of alternate angles, then x° = ......................

b) If a° and 120° are a pair of corresponding angles, then a° = ......................

c) If p° and q° are a pair of co-interior angles, then p° + q° = ......................

d) If x° and 40° are a pair of co-interior angles, then x° = ......................

e) If m° and 150° are a pair of alternate exterior angles, then m° = ......................

Creative Section - A

4. a) What is a transversal? Write with a diagram.

b) Define alternate angles with the help of a diagram.

c) Define corresponding angles with the help of a diagram.
d) Define co-interior angles with the help of a diagram.
e) Write the properties of alternate angles, corresponding angles and co-interior

angles made by a transversal between parallel lines.

5. Let's name the pairs of alternate angles, corresponding angles and co-interior
angles in the following figures.

a) A b) P c) d) C
B
PQ Z YD
BC W
X Y XT A
Q R

6. Let's find the sizes of unknown angles.

a) b) c) p° d) c°

y° b° 60° q° b° d°
x° c° r° a°
40° a°
45° 65°
e°

229Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 7

Geometry: Angles

e) f) f° g) w° h)

110° w° 105° b° c° 85° y°
z° y° x°
e° 110° x° y° x° 130°
70° a° d° z°

i) c° j) k) l)

145° z° y° c° b° 55°
x° 115°
a° b° a° 150° a°
65° b°

m) n) 100q o) p) w°
w°
y° z° z° a° x° z° 55°
x° 80° x° y° 35°
y° x° y° 65°
25° b°

7. From the given figure, let's find the sizes of unknown angles .

a) b) c) d)

2x° (2x 25)q (3a 5)q (2x 69)q (x 96)q
(2a 5)q
y° x° (x 55)q

Creative Section - B

8. In the given figures, find the measurements of unknown angles.
a) b) c) d)

35° 150° 140° 140°
100°
x° x° y° x°
25° 20° 130°

9. a) In the figure alongside, show that w° x° a° b°
(i) ‘w = ‘c (ii) ‘x = ‘s (iii) ‘y = ‘g z° y° d° c°
(iv) ‘a = ‘r (v) ‘d = ‘q (vi) ‘p = ‘c p° q° e° f°
s° r° h° g°

b) In the adjoining figure, show that x° y°
‘a + ‘b + ‘c = 2 right angles. a°

b° c°

c) In the given figure, show that ‘a + ‘b + ‘c + ‘d = 360q. y° c° b°

d° a° x°

Vedanta Excel in Mathematics - Book 7 230 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Geometry: Angles

d) In the given figure, show that y = 120°. y°
x°
60°

e) In the given figure, show that ‘a + ‘b = ‘c + ‘d b° a°
d°
f) In the figure alongside, find ‘x and ‘y, and
show that BF and CE are parallel to each other. B c°

10. a) In the given figure, AB // DE, ‘BAC = 120° and F E
‘CDE = 150°, find the value of x° A 100°
[Hint: Produce ED to intersect AC at F.]
x° y°
b) From the figure given alongside, find the value of x. 45° C 55°D
[Hint: Through C, draw CZ // DY // BX]
A B
E
120°
FD

150°

CA X
B 3x°

C 90° Y

D 2x°
E

m°

c) In the figure, if m : n = 4 : 5, find the measure of ‘p. p°
n°

It's your time - Project work!

11. a) Let's cut three sets of 3 paper strips from a chart paper. From each set of 3
paper strips make the models of alternate angles ( -shaped), corresponding
angles ( -shaped), and co-interior angles ( -shaped) between two parallel
strips intersected by another strip using glue stick.

b) With the help of a ruler and set squares, let's draw three pairs of parallel lines
separately. Intersect each pair of parallel lines by a transversal. Measure all
angles made by the transversal between each pair of parallel lines with the
help of a protractor. Then, explore the relationships between the following
pairs of angles.

(i) Relationship between alternate angles

(ii) Relationship between corresponding angles

(iii) Relationship between co-interior angles

231Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 7

Unit Triangle, Quadrilateral and Polygon

14

14.1 Triangles – Looking back
Classroom - Exercise

Let's tell and write the types of triangles selecting from the following types.
scalene triangle, isosceles triangle, equilateral triangle, acute-angled triangle,
obtuse-angled triangle, right-angled triangle.

1. a) A triangle which has two equal sides. ...................................................
b) A triangle with none of the equal sides. ...................................................
c) A triangle which has all three equal sides. ...................................................
...................................................
2. a) A triangle which has a right angle. ...................................................
b) A triangle which has an obtuse angle. ...................................................
c) A triangle with three acute angles.

A triangle is a closed plane figures bounded by three line segments. A C
It has three sides, three vertices, and three angles. In the given B
triangle ABC, AB, BC and CA are it's three sides, A, B, and C are
three vertices. ‘A, ‘B and ‘C are three angles of 'ABC.

14.2 Properties of triangles

Property 1 P
R
The sum of the length of any two sides of a triangle is always

greater than the length of its third side.

In 'PQR, Q
PQ + QR > PR, QR + PR > PQ and PQ + PR > QR.

Property 2 P

The angle opposite to the longest side of a triangle is the 2.5 cm 5 cm R
greatest in size and the angle opposite to the shortest side Q 4 cm
is the smallest in size.

In 'PQR, the longest side is PR and its opposite angle is ‘Q.
The shortest side is PQ and its opposite angle is ‘R.

So, ‘ Q is the greatest and ‘R is the smallest angles of 'PQR.

Vedanta Excel in Mathematics - Book 7 232 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Triangle, Quadrilateral and Polygon

Conversely, the side opposite to the greatest angle of a triangle is the longest one
and the side opposite to the smallest angle is the shortest one.

Property 3 P

The sum of the angles of a triangle is always 180q. Q R
In 'PQR, ‘P + ‘Q+ ‘R = 180q P

Property 4

In an isosceles triangle, the angles opposite to the equal sides are
always equal.

In the figure, sides PQ = PR, So, ‘Q = ‘R Q R

Such equal angles of an isosceles triangle are also called base angles.

Property 5 P
60°
All three sides and angles of an equilateral triangle are equal
and the size of each angle is 60°. 60° 60°
QR

In the figure, 'PQR is an equilateral triangle in which PQ = QR = PR and

‘P = ‘Q = ‘R = 60°.

Property 6 P

The exterior angle of a triangle is equal to the sum of its two Q RS
opposite interior angles.
In 'PQR, ‘PRS = ‘PQR + ‘QPR

Worked-out examples

Example 1: If the angles of a triangle are in the ratio 2 : 3 : 4, find them.
Solution:
Let the angles of the triangle be 2xq, 3xq and 4xq respectively.

Now, 2xq + 3xq + 4xq = 180q [Being the sum of the angles of a triangle.]

or, 9xq = 180q

or, xq = 180°
9

= 20q.

? 2xq = 2 u 20 = 40q, 3xq = 3 u 20q = 60q and 4xq = 4 u 20q = 80q.

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 233 Vedanta Excel in Mathematics - Book 7

Triangle, Quadrilateral and Polygon

Example 2: In the given right-angled triangle, find the size of 3x°
each acute angle.
2x°
Solution:

In a right-angled triangle, the sum of its two acute angles is always 90°.

? 2x° + 3x° = 90

Now, 5xq = 90q

or, xq = 90° = 18q
5

Now, 2xq = 2 u 18° = 36q, 3xq = 3 u 18q = 54q

Hence, the required acute angles are 36° and 54°.

Example 3: Find the sizes of unknown angles in the following figures.

a) P b) A

z 84° D

Ba F 48° C
b25°
y x 110° E
Q RS

Solution:

a) x + 110q = 180q [Being the sum a straight angle]
or, x
y = 180q – 110q = 70q
Again, y + z
= x = 70q [Being the base angles of an isosceles triangle]
or, 70q + z
or, z = 110q [Being the sum equal to exterior angle of the

triangle]

= 110q

= 110q – 70q = 40q

? x = y = 70q and z = 40q.

b) a + 48q + 84q = 180q [Being the sum of the angles of ' ABC]

or, a = 180q – 132q = 48q

Again, b + 25q = a [Being the sum equal to exterior angle of 'BEF]

or, b + 25q = 48q

or, b = 48q – 25q = 23q

?a = 48q and b = 23q.

EXERCISE 14.1

General Section - Classwork

1. a) Write the side opposite to the vertex A of ∆ABC. .............................

b) In 'PQR, name the vertex opposite to the side PQ. .............................

c) Name the angle opposite to side XZ in 'XYZ .............................

d) Write down the name of side opposite to ‘ABC of 'ABC. .............................

Vedanta Excel in Mathematics - Book 7 234 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Triangle, Quadrilateral and Polygon

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

b 6.5 cm Q D

R

greatest angle is ......... EF
smallest angle is .........
P
dX
greatest angle is ......... greatest angle is .........
smallest angle is ......... smallest angle is .........

eK L C

YZ M

AB

longest side is ......... longest side is ......... longest side is .........

shortest side is ......... shortest side is ......... shortest side is .........

3. In the given triangles let's tell and write the size of unknown angles as quickly
as possible.

60° 50°

xq 40° zq

x° = .............. y° = .............. z° = ..............

4. Let's tell and write 'True' or 'False' in the blanks.

(a) It is possible to construct a triangle with sides 5cm, 4cm and 10cm? ...............
(b) We can have a triangle with angles 40°, 60°, and 80°. ........................

(c) If one of the acute angles of a right-angled triangle is 50°, then the other

acute angle is 30° ........................

(d) If two opposite interior angles of an exterior angle of a triangle are 50° and

70°, then the size of the exterior angle 120° ........................
Creative section - A

5. a) Write the types of triangle according to the length of sides of triangles.
b) Write the types of triangle according to the size of angles of triangles.
c) What are the base angles of an isosceles triangle? Write with a diagram.
d) Write the relation between base angles of an isosceles triangle.

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 235 Vedanta Excel in Mathematics - Book 7

Triangle, Quadrilateral and Polygon

e) Write the relation between the exterior and opposite interior angles of a
triangle.

6. a) In a triangle ABC, AB = 5.5cm, BC = 4cm and CA = 3.8cm. Sketch the
triangle and name the greatest and the smallest angles of the triangle.

b) In a triangle PQR, ‘P = 110°, ‘Q = 30° and ‘R = 40°. Sketch the triangle
and name the longest and the shortest sides of the triangle.

7. a) If one of the acute angles of a right-angled triangle is 27°, find the other acute
angle of the triangle.

b) An acute angle of a right-angled triangle is twice the other acute angle. Find
them.

c) If two acute angle of a right-angled triangle are 2x° and 3x° respectively, find
the size of each acute angle.

d) If x°, 2x° and 60° are the angles of a triangle, find the sizes of unknown
angles.

e) If the angles of a triangle are in the ratio 4 : 5 : 6, find them.

8. Let's find the sizes of unknown angles in the following figures.

a) b) 3y° 36° c) d) x°

x° y° a° 2x°
3x°
60° 45° (a+10)°

e) f) y° 30° g) 3x° 100° h)
66°
70° 120° x° 54°
x°
m° n°

i) j) k) l) x° y° z°
40°
y° c° x° z°
(x+15)° 130°
b° a° 110° y°
x°

9. Let's find the sizes of unknown angles in the following figures.

a) b) c) d) z° y° 64°
x°
35° 30° z° 44° 50° 48°
y° 62° a°
x° z° y°
x° 56° b° c°

Vedanta Excel in Mathematics - Book 7 236 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Triangle, Quadrilateral and Polygon

e) f) g) h)

58° d° c° x° 95° z° a° 78° 2a°
a° b° 50° x° 55°
y° b°
y° z°

Creative section - B xy
a
10. a) In the given figure, using the property of alternate angles
between parallel lines and the property of a straight angle, show bc
that: a + ‘b + ‘c = 180°.

b) In the adjoining figure, show that ‘x = ‘a + ‘b. a x
bc

c) In the figure alongside, show that, ‘x is a right angle. a
a

x
bb

A

11. a) In the given ∆ABC; AB = BC and BD = CD. Find the value c°
of a°, b° and c°. D

a° b° C
B D

A

b) Using the properties of alternate angles and corresponding a
angles made by a transversal between parallel lines in b c xy

the given figure, show that ‘ACD = ‘a + ‘b. B C
P

c) In the given figure, 'PQR is an isosceles triangle in which

PQ = PR. Prove that ‘Q = 90° – ‘P . Q R
2

It's your time - Project work!

12. a) Let's take three straws (Plastic tubes which are used to suck cold drinks, juice,
etc. from glass or bottle) of different lengths such that the sum of the lengths
of two straws is greater than the remaining straw. Join them together using
cello tape to form a model of triangle. Write a short report about the relation
between the longest side, short sides, greatest angle and the smallest angle.
Then present your report in the class.

b) Let's take three straws such that the sum of the length of two straws is shorter
than the remaining straw. Now, make a model of a triangle using these straws.
Is it possible to make a triangle? Write a short report on it and present in the
class.

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 237 Vedanta Excel in Mathematics - Book 7

Triangle, Quadrilateral and Polygon

13. a) Let's draw a triangle of your own choice of measurements. Measure each
angle of your triangle and find their sum. Present your finding in the class.

b) Let's take a square sheet of paper and fold it through it's one diagonal to form
a triangle. Measure thee sides of this triangle. What types of triangle is it?
Measure the size of angles opposite to the equal sides of this triangle just
by overlapping one angle on the other. Are these angles equal? Present your
findings in the class.

c) Let's draw a triangle of your own choice of measurements. Produce one side of
your triangle to make an exterior angle. Measure the size of exterior angle and
it's two opposite interior angles using a protractor. Now, write a short report
about the relationship between exterior angle of a triangle and it's opposite

interior angles and present in the class.

14.3 Some special types of quadrilaterals

Quadrilaterals are the polygons having four sides. Parallelogram, rectangle, square,
rhombus, trapezium and kite have some special properties. So, they are called
special types of quadrilaterals.

(i) Parallelogram

Its opposite sides are equal and parallel. D C
? AB = DC and AB // DC, AD = BC and AD // BC

Its opposite angles are equal. AB
? ‘A = ‘C and ‘B = ‘D DC
Its diagonals bisect each other.

? Diagonals AC and BD bisect each other at O. O
i.e. AO = OC and BO = OD. AB

(ii) Rectangle DC
Its opposite sides are equal and parallel.

? AB = DC and AB // DC, AD = BC, and AD // BC. A B
Its all angles are equal and each of them is 90q.

‘A = ‘B = ‘C = ‘D = 90q DC

Its diagonals are equal and they bisect each other. O C
? Diagonals AC = BD, AO = OC, and BO = OD. CB
B
(iii) Square D C
Its all sides are equal
B
? AB = BC = CD = DA A 90°
Its all angles are equal and they are 90q

? ‘A = ‘B = ‘C = ‘D = 90q D

Its diagonals are equal and they bisect each other at right angle. O
A
? AC = BD and they bisect each other at O at right angle.
i.e., AO = OC, BO = OD, and BD A AC at O.

Vedanta Excel in Mathematics - Book 7 238 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Triangle, Quadrilateral and Polygon

(iv) Rhombus B C
Its all sides are equal and opposite sides are parallel. A

? AB = BC = CD = DA and AB // DC, AD // BC

Its opposite angles are equal. D

? ‘A = ‘C and ‘B = ‘D B

Its diagonals are not equal but they bisect each other at right angle. AO C

?AC and BD bisect each other at O at right angle,

i.e., AO = OC, BO = OD and BD A AC at O. D
DC
(v) Trapezium
Its any one pair of opposite sides are parallel. A A C B
? AB // DC. B C

(vi) Kite D
Its particular pairs of adjacent sides are equal. B
AB = AD and BC = DC
The opposite angles formed by each pair of AO
unequal adjacent sides are equal. D
‘ABC = ‘ADC.
Diagonals intersect each other at right angle.
AC and BD intersect at O at right angle.
i.e. BD A AC at O.

14.4 Verification of properties of special types of quadrilaterals

A. Verification of properties of parallelogram

Activity 1 – Classwork
Verify experimentally that the opposite angles of parallelogram are equal.

(i) Measure the opposite angles of the following parallelograms ABCD with a

protractor. AD A

DC

B
D

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

(ii) Write the measurements in the table.
Fig. No. Measurement of opposite Measurement of opposite

angles angles Result

(i) ‘ABC ‘ADC ‘BAD ‘BCD ‘ABC = ‘ADC
‘BAD = ‘BCD
(ii)
(iii)

Conclusion: The opposite angles of parallelogram are equal.

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 239 Vedanta Excel in Mathematics - Book 7

Triangle, Quadrilateral and Polygon

Activity 2 – Classwork
Verify experimentally that the opposite sides of parallelogram are equal.

(i) Measure the opposite sides of the following parallelograms PQRS with a

ruler R P Q Q

S

P
R

.P Fig (i) Q S R Fig (iii) S
Fig (ii)

(ii) Write the measurements in the table.

Fig. No. Measurement of opposite Measurement of opposite

sides sides Result

PQ SR PS QR PQ = SR
(i) PS = QR
(ii)

(iii)

Conclusion: The opposite angles of parallelogram are equal.

Activity 3 – Classwork
Experimentally verify that the diagonals of parallelogram bisect each other.

(i) Measure the lengths of the intersected parts AO, OC and BO, OD of the

diagonals with a ruler in each parallelogram ABCD.
DC
AD C B

O O C O
A FIG (I) B
B DA
FIG (II) FIG (III)

(ii) Write the measurements in the table.

Fig. No. Lengths of the parts of Length of the parts of Result
diagonal AC diagonal BD
AO = OC
AO OC BO OD BO = OD

(i)

(ii)

(iii)

Conclusion: The diagonals of parallelogram bisect each other.

Vedanta Excel in Mathematics - Book 7 240 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Triangle, Quadrilateral and Polygon

B. Verification of properties of rectangle

Activity 4 – Classwork
Experimentally verify that the diagonals of rectangle are equal.

(i) Measure the length of diagonals PR and QS with a ruler in each of the

following rectangle PQRS. Q

S RP S

P

R

P QQ R
Fig (i)
Fig (ii) Fig (iii) S

(ii) Write the measurements in the table.

Fig. Length of diagonals Result
PR QS PR = QS
No.

(i)
(ii)
(iii)

Conclusion: The diagonals of rectangle are equal.

C. Verification of properties of square

Activity 5 – Classwork
Verify experimentally that the diagonals of square bisect each other perpendicularly.

(i) Measure the length of the parts of the diagonals and an angle made by them

at the point of intersection in each square ABCD. A

CB

DC D

OO
OB

AB D A C
Fig (i) Fig (ii) Fig (iii)

ii) Write the measurements in the table.

Fig. Length of the parts of

No. diagonals AC and BD ∠AOB Result
AO OC BO OD

(i) AO = OC = BO = OD
(ii)
(iii) ∠AOB = ∠BOC = ∠COD = ∠DOA = 90°

Conclusion: The diagonals of square bisect each other perpendicularly.

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 241 Vedanta Excel in Mathematics - Book 7

Triangle, Quadrilateral and Polygon

Activity 6 – Classwork
Verify experimentally that the diagonals of square bisect the vertical angles.

(i) Measure the parts of each vertical angles of the following squares PQRS with a

protractor P

PS

SR Q

OO
OS

PQ QR R
Fig (i) Fig (ii) Fig (iii)

(ii) Write the measurements in the table.

Fig. Parts of ∠QPS Parts of ∠PQR Parts of ∠QRS Parts of ∠PSR Result

No. ∠QPO ∠SPO ∠PQO ∠RQO ∠QRO ∠SRO ∠RSO ∠PSO ∠QPO = ∠SPO

(i) ∠PQO = ∠RQO
(ii) ∠QRO = ∠SRO

(iii) ∠RSO = ∠PSO

Conclusion: The diagonals of square bisect the vertical angles.

D. Verification of properties of rhombus
Activity 7 – Classwork

Experimentally verify that the diagonals of rhombus bisect each other
perpendicularly.

(i) Measure the length of the parts of the diagonals and an angle made by them

at the point of intersection in each rhombus ABCD.

D CA B

CD

OO O

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

(ii) Write the measurements in the table.

Fig. Length of the parts of

No. diagonals AC and BD ∠AOB Result
AO OC BO OD

(i) AO = OC

(ii) BO = OD

(iii) ∠AOB = ∠BOC = ∠COD = ∠DOA = 90°

Conclusion: Diagonals of rhombus bisect each other perpendicularly.

Vedanta Excel in Mathematics - Book 7 242 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Triangle, Quadrilateral and Polygon

Activity 8 – Classwork
Verify experimentally that the diagonals of rhombus bisect the vertical angles.

(i) Measure the vertical angles ∠A, ∠B, ∠C and ∠D and the parts of each vertical
angle made by diagonals in each of the following rhombus.

D CA B D
C

O O O
AB C
B A
Fig (iii)
Fig (i) D Fig (ii)

(ii) Write the measurements in the table.

Fig. Vertical Parts of ∠A Parts of ∠B Parts of ∠C Parts of ∠D Result
No. angles

∠A ∠B ∠C ∠D ∠BAO ∠DAO ∠ABO ∠CBO ∠BCO ∠DCO ∠ADO ∠CDO ∠BAO=∠DAO=1/2∠A
(i) ∠ABO=∠CBO= 1/2∠B

∠BCO=∠DCO= 1/2∠C
(ii) ∠ADO=∠CDO=1/2∠D
(iii)

Conclusion: The diagonals of rhombus bisect the vertical angles.

Worked-out examples

Example 1: From the given figures, find the value of x.

a) D (x )cm C b) S R c) A (2x )cm (x 3)cm D
120q

(2x )° B C
P
A (3x )cm B Q

Solution:

a) Here, AB = CD [ Being opposite sides of parallelogram ]
=x+1
or, 3x – 5

or, 3x – x =1+5

or, 2x =6

or, x = 6 = 3
?x 2

=3

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 243 Vedanta Excel in Mathematics - Book 7

Triangle, Quadrilateral and Polygon

b) Here, ‘P = ‘ R [Being opposite angles of parallelogram ]
or, (2x + 30)°
or, = 120°
or, 2x°
or, 2x° = 120° – 30°

? x° = 90°
90°
x° = 2 = 45°

= 45°

c) Here, AC = BD [Diagonals of a rectangle are equal]

or, 2x – 1 = x + 3
or, 2x – x = 3 + 1
or, x = 4

?x =4

Example 2: If xq, (x + 20)q (x + 40)q and (x + 60)q are the angles of a quadrilateral,

find them.

Solution:

Here, x° (x + 20)° (x + 40)° (x + 60)° = 360q [Sum of the angles of quadrilateral]

or, (4x + 120)q = 360q

or, 4xq = 360q – 120q

or, x = 240° = 60q
4
? xq = 60q , (x + 20)q q q q (x + 40)q q q q

and (x + 60)q q q q

Example 3: If the angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4, find them.
Solution:
Let the angles of the quadrilateral be xq, 2xq, 3xq and 4xq respectively.

Here, xq + 2xq + 3xq + 4xq = 360q
or, 10xq = 360q

or, xq = 360° = 36q
10

? x q = 36q, 2xq = 2 u 36q = 72q, 3xq = 3 u 36q = 108q and 4xq = 4 u 36q = 144q.

Example 4: Find the unknown sizes of angles in the following figures.

a) b) x°

2x°+10° 2x° y°

x° y° 105° w° 20° z°

Solution: [Being the sum a straight angle]
a) yq + 105q = 180q

or, yq= 180q – 105q = 75q

Vedanta Excel in Mathematics - Book 7 244 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Triangle, Quadrilateral and Polygon

Now,xq + (2xq + 10q) + 2xq + yq = 360q [The sum of the angles of a quadrilateral]

or, 5xq + 10q + 75q = 360q

or, 5xq = 360q – 85q

or, xq = 275° = 55q
5

? xq = 55q, 2xq + 10q = 2 u 55q + 10q = 120q, 2xq = 2 u 55q = 110q and yq = 75q.

b) Here, wq = 25q [Being alternate angles]

xq = 20q [Being alternate angles]

Now, wq + 20q = 25q + 20q = 45q

xq + 25q = 20q + 25q = 45q

Again, yq + 45q = 180q [Being the sum of co-interior angles]
or, yq = 180q – 45q [Being the opposite angles of a parallelogram]
or, yq = 135q
Also, zq = yq = 135q

?wq = 25q, xq = 20q, yq = zq = 135q.

EXERCISE 14.2
General Section - Classwork
1. Let's tell and write true or false for the following statements.

a) The opposite sides of a parallelogram are equal .....................
b) The opposite angles of a parallelogram are unequal .....................
c) Every rectangle is a parallelogram. .....................
d) Every parallelogram is a square .....................
e) The diagonals of rectangle do bisect to each other .....................
f) The diagonals of a square and a rhombus bisect each other

perpendicularly .................
g) If an angle of a parallelogram is 90°, it becomes a rectangle .....................
h) The diagonals of a kite bisect to each other. .....................
2. Let's tell and write the correct answers as quickly as possible.

(a) If w°, x°, y°, and z° are the angles of a quadrilateral, then
w° + x° + y° + z° = ..................

(b) If x° represents an angle of a rectangle, then x° = ..........................

(c) If a° and 115° are the opposite angles of a parallelogram, then
a° = ..........................

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 245 Vedanta Excel in Mathematics - Book 7

Triangle, Quadrilateral and Polygon

(d) If p° and 70° are the opposite angles of a rhombus, then p° = ..........................

(e) If b° is the angle formed by the intersection of the diagonals of a square,
then b° = ........................

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

a) b° a° b) c) 65° x°

130°

55° x° 60° y°

a° = .......... x° = .......... x° = ..........
b° = .......... y° = ..........

Creative Section - A

4. a) Are all rectangles parallelogram? Justify your answer.

b) Are all parallelograms rectangle? Justify your answer.

c) Are all squares rectangle? Write with reasons.

d) Are all rectangles squares? Write with reasons.

e) Are all squares rhombus? Give reasons.

f) Are all rhombuses square? Give reasons.

g) Write the opposite angles and opposite sides properties of a parallelogram.

h) Write the diagonal properties of: (i) a rectangle (ii) a square(ii) a rhombus
(iv) a kite

5. Let's find the value of x in each of the following figures.

a) b) c) d)

A (2x + 1) cm D P S P O W Z
70° (4x – 15)°
(3x – 2) cm
(x + 4) cm 2x° NX 105°
Y
B (x + 5) cm C Q RM
C
e) f) g) h)

D 5 cm P OS RX W

(x – 3) cm (x+4)cm 7cm (3x–5()xc+m5)cm
(7– x) cm
P QU V
A (2x – 3) cm B M N

Vedanta Excel in Mathematics - Book 7 246 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Triangle, Quadrilateral and Polygon

6. Let's work out these problems.
a) If aq, 60q, 120q and 130q are the angles of a quadrilateral, find aq.
b) If xq, 2xq, 85,q and 140q are the angles of a quadrilateral, find xq and 2xq.
c) If aq, 2aq, 4aq and 5aq are the angles of a quadrilateral, find them.
d) If xq, 3xq, (xq + 15q), and (2xq + 30q) are the angles of a quadrilateral, find
them.
e) If the angles of a quadrilateral are in the ratio 2 : 3 : 4 : 6, find them..

7. Let's find the unknown sizes of angles in the following figures.

a) b) c) d)

90° 105° a° 2a° 3x° p° 4p°
x° (x+13)°
2x° x° (a+10)° 2p°+10°
(x+5)° 3p°

e) y° f) b° c° g) 100° h) x°
y° x°
z° 110° a° 4c° 130°
z° z°+10° 2xq
2z° x° 80°
y°
60° z°

Creative Section - B

8. Let's find the unknown sizes of angles in the following figures

a) b) c) 50° d) x°20°
z°
y° x° c° b° p° w°
z° a° 110° r° s° y° 15°
60° d° q°

e) f) g) h)

x° w° r° s° c° a° x° x°
z° 88°
q° y° z° 32°
60° 30° y°
65° y° p° x°y° b°

9. a) Let's calculate the unknown sizes of the angles of the x° 2y°
adjoining trapezium. 50° y°

b) Let's calculate the unknown sizes of the angles of the x° 120° w°
adjoining parallelogram.
25° y°
c) Find the unknown sizes of the angles of the given kite.
20° x° y°
42°
d° a° b°c°

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 247 Vedanta Excel in Mathematics - Book 7

Triangle, Quadrilateral and Polygon

B D TK
55°
10. a) In the figure alongside, BEST is a rectangle and a°
DESK is a parallelogram. If ‘SKD = 55° and E S
‘BED = a°, find the measure of a°. ED
A
68°

b) In the given parallelogram ABCD, if AB = AE, find y°

the value of y°. B C
DH
AE

c) In the adjoining figure, ABCD and EFGH are 50°

parallelogram . If ‘ABC = 60°, ‘EOD = 50°, and 60° x°
G
‘FGH = x°, find the measure of x°. B CF

It's your time - Project work

11. a) Let's take a rectangular sheet of paper and fold it
through it's two diagonals.

(i) Do the diagonals bisect each other?

(ii) Are two diagonals equal?

(iii) Do the diagonals intersect each other
perpendicularly?

(iv) Write a short report about your findings and present in the class.

b) Let's take a square sheet of paper and fold it through its two
diagonals.

(i) Are the diagonals equal?

(ii) Are two diagonals equal?

(iii) Do the diagonals intersect each other perpendicularly?

(iv) Write a short report about your findings and present in
the class.

14.5 Interior and exterior angles of regular polygons

A polygon is a closed plane figure bounded by three or more line segments.
Triangle, quadrilateral, pentagon, hexagon, etc. are the examples of polygons.

Name of Figure No. of No. of Sum of Rules
Polygon sides triangles angles

Triangle 3 1 1 × 180° (3 – 2) × 180°

Vedanta Excel in Mathematics - Book 7 248 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur