Vedanta Excel in Mathematics Book 7 Final (2079)

Coordinates

Object Image Object Image

Line AB Line PQ ∠GHI ∠UVW

Line GH ................. ∠CED .................

Line DE ................. ∠RST .................

Line ST ................. ∆CDE .................

2. Let's say 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) → .......................... P (x, y) → ..........................
P (–x,y) → .......................... P (–x,y) → ..........................
P (–x, –y ) → .......................... P (–x, –y ) → ..........................
P (x, –y) → .......................... P (x, –y) → ..........................
A ( 3, 4) → .......................... A (2, 5) → ..........................
B (−5, 2) → .......................... B (−3, 6) → ..........................
C (–6, –1) → .......................... C (–1, –4) → ..........................
D (4, –3) → .......................... D (6, –7) → ..........................

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. M

a) A b) M Q N c) D A

B C R
M NP C

NB

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Coordinates

d e) f)

M U N MN M
P 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

66 6
5 E 5 5
4 A R 4 4
F 3
3 Q3
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--021 1 2Y3 4 5 6 X
-2 I H -3 Z
-2 -3
-3
-4 -4 -4
-5 -5 -5 X
-6 -6 G L -6 W

Y' Y' Y'

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

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.

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Coordinates

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".

10. a) ∆ 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.

b) 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.

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13.3 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.

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

P' O P
90°

90° P P'
O
Point P is said to be rotated through 90°
Point P is said to be rotated through 90° 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.

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Coordinates

P' O P

180° 180°

P' O P Point P is said to be rotated through 180°
Point P is said to be rotated through 180° 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 90° in anti-clockwise direction

(i) Join each vertex of the figure to the centre of C' B' Anti-clockwise
rotation with dotted lines.
C
(ii) On each dotted line draw 90° at O with the help A'
of a protractor in anti-clockwise direction.
B
(iii) With the help of compasses, cut off OA' = OA, O
OB' = OB and OC' = OC.
A

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

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

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Coordinates

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

given centre of rotation.

B’

13.4 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 90° and 180° 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'(3,-2)
P'(-1,-3) P(-2,-3)

Y' Y' Y' Y'

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

∴ P (x, y) → P' (–y, x) ∴ P (–x, y) → P' (–y, –x) ∴ P (–x, –y) → P' (y, –x) ∴ P (x, –y) → P' (y, x)

Thus, when a point is rotated through 90° 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) → P' (– y, x)

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

∴ P (x, y) → P' (y, –x) ∴ P (–x, y) → P'(y, x) ∴ P (–x, –y) → P' (–y, x) ∴ P (x, –y) → P' (–y, –x)

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Coordinates

In this way, when a point is rotated through 90° 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) → P' (y, – x)

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

When a point is rotated through 180° 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(-3,1) P'(2,3)
X X' OX
X' O X X' O X X' O P(2,-2)
P'(-3,-2) P'(3,-1) P(-2,-3)
Y'
Y' Y' Y'

P (3, 2) → P" (–3, –2) P (–3, 1) → P' (3, –1) P (–2, –3) → P' (2, 3) P (2, –2) → P' (–2, 2)

∴ P (x, y) → P' (–x, –y) ∴ P (–x, y) → P' (x, –y) ∴ P (–x, –y) → P' (x, y) ∴ P (x, –y) → P' (–x, y)

Thus, when a point is rotated through 180° 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) → P' (– x, – y)

EXERCISE 13.2
General Section - Classwork

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

Clockwise rotation through 90° Anti-clock wise rotation through 90°

a) P (x, y) → ....................... a) P (x, y) → .......................

b) P (–x, y) → ....................... b) P (–x, y) → .......................

c) P (– x, –y) → ....................... c) P (–x, –y) → .......................

d) P (x, –y) → ....................... d) P (x, –y) → .......................

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e) A(–2, 3) → ....................... Coordinates
e) B (4, –6) → .......................

f) C (5, 2) → ....................... f) (–3, –1) → .......................

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

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

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

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

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

Creative Section - A

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

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

H O

A O EF
R
O

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 90° in anti-clockwise direction. Also, write the coordinates of
the vertices of images.

YYY
6 6 6
a) 5 E b) F 5 c) 5 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--32
-2 -2
-3 -3
-4 -4 -4
-5 -5 LM -5
-6 -6 K -6

Y' Y' Y'

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Coordinates

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

rotating through 180° in anti-clockwise direction. Also write the coordinates
of the vertices of images.

a) 6Y b) 6Y E c) Y
5 6
K L 5 Y P A D 5
4 B 4
3 4 3
3
222
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 2 345 6
S R--23 W
-2 -2
-3 -3
-4 -4 -4
-5 -5 T -5 Y 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 90° 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 90° 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 180° 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 180° 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.

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.

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Coordinates

13.5 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

A C' B

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

Worked-out examples

Q

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

Solution:

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

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

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

(iv) Join P' and Q'. Y
P'Q' is the required image of the line PQ. P

P'

A

Example 2: Displace the triangle ABC in the magnitude and O

direction of OP. C B
A
Solution: OP

(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) DA'B'C' is the required image of DABC under the displacement of the magnitude
and direction of OP.

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Coordinates

EXERCISE 13.3

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)

PQ P P
E
A B Q ZD
Q
d) S R e) Y f) A

X C S
P

O

AO P R
PQ O

X BQ

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

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Coordinates

(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.

OBJECTIVE QUESTIONS

Let’s tick (√) the correct alternative.
1. The coordinates of the origin are ...................

(A) (0, 1) (B) (2, 0) (C) (0, 0) (D) (-3, 3)

2. A point is 4 units right from the origin along x–axis and 9 units down along y-axis, then
its coordinates are …………………

(A) (4, 9) (B) (-4, 9) (C) (-4, -9) (D) (4, -9)

3. A point is 7 units left from the origin along x–axis and 3 units up along y-axis, then its
coordinates are ………………

(A) (7, 3) (B) (-7, 3) (C) (7, -3) (D) (-7,-3)

4. If the abscissa of a point is –8 and its ordinate is 5, then its coordinates are …………

(A) (-8, 5) (B) (8, -5) (C) (-8, -5) (D) (8, 5)

5. In which quadrant does the point (-6, 1) lie?

(A) First quadrant (B) Second quadrant (C) Third quadrant (D) Fourth quadrant

6. The coordinates of image of the point (x, y) after reflection on x-axis are

(A) (x, y) (B) (x, -y) (C) (-x, -y) (D) (-x,-y)

7. What will be the image of a point (2, 3) under the reflection about x-axis?

(A) (2, 3) (B) (2, -3) (C) (-2, 3) (D) (-2, -3)

8. The coordinates of image of the point (x, y) after reflection on y-axis are

(A) (x, y) (B) (x, -y) (C) (-x, -y) (D) (-x,-y)

9. What will be the image of a point (-7, 4) under the reflection about y-axis?

(A) (7, 4) (B) (7, -4) (C) (-7, 4) (D) (-7, -4)

10. A point P lies above x-axis, where will its image lie after reflection about –axis?

(A) upwards x-axis (B) downwards axis (C) right to y-axis (D) left to y-axis

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Unit 14 Geometry: Lines and Angles

14.1 Lines and Angels – Looking back

Classwork - Exercise

1. Let's say and write the names of pair of intersecting, parallel and non-parallel
line segments.
QS
K WY
A
EP
F QM N

C BD G HR SL X ZR T





Intersecting Line Segments Parallel Line Segments Non-parallel Line Segments

................ and ................ ............ and ............ ................ and ................
................ and ................ ............ and ............ ................ and ................

2. Let's say and write the names, vertices and arms of these angles.

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

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

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

3. Let's say 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 e) A C f) O 300° G

125°

O T D
M

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

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Geometry: Lines and Angles
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.

14.2 Construction of different angles DSC
Construct 60°, 120°, and 90° with the help of compasses. RQ
Steps of construction

(i) Draw a line segment AB. AP B

(ii) With centre at A and a suitable radius, draw an arc to cut AB at P.

(iii) From P, draw an arc with the same radius to cut the first arc at Q. Join A, Q, and
produce to C.

Here, ∠BAC = 60°.

(iv) From Q draw an arc with the same radius to cut the first arc at R. Join A, R, and
produce to D.

Here, ∠BAD = 120°.

(v) From Q and R draw two arcs with the same radius to intersect each other at S.

(vi) Join A and S.

Here, ∠BAS = 90°. D E C
Construct 75° with the help of compasses. R

Steps of construction QP B
(i) Draw a line segment AB. A
(ii) At A, construct ∠BAC = 60° and ∠BAD = 90°.

(iii) From P and Q, draw two arcs with the same radius to intersect each other at R.

(iv) Join A, R and produce to E.

Here, ∠BAE = 75°.

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Geometry: Lines and Angles

Construct 30° and bisect it. C

Steps of construction Q RD
S TE
(i) Draw a line segment AB.
AP B
(ii) At A, construct ∠BAC = 60°.

(iii) From P and Q, draw two arcs to intersect each other at R.

(iv) Join A, R and produce to D.

Here, ∠BAD = 30°.

(v) From P and S draw two arcs to intersect each other at T.

(vi) Join A, T and produce to E.

Here, ∠BAE = 1 ∠BAD = 1 × 30° = 15°.
2 2

14.3 Construction of equal angle using compass

The process of constructing an angle equal to B R
T
the given angle at the given vertex is called

transferring angles. D

Construct an angle at Q equal to ∠AOB. OC A QS P
Steps of construction

(i) From the point Q, draw a line segment QP.

(ii) With centre at O and a suitable radius draw an arc to intersect OA at C and OB
at D.

(iii) With centre at Q, draw the same arc to intersect QP at S.

(iv) Measure the length of CD with the help of compasses and draw an arc of the
same length from S to cut previous arc at T.

(v) Join Q and T and produce it to R.

Thus ∠PQR = ∠AOB is drawn.

EXERCISE 14.1

1. Let's construct the following angles with the help of compasses.

a) 60° b) 120° c) 90° d) 30° e) 45°
f) 75° g) 105° h) 150° i) 15°
j) 22 1 °
2

2. Let's construct an angle equal to the given angle at the given vertices.

a) C b) Z S c) R L
B
P X YQ P
A

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3. Let's draw the following line segments in your exercise book and draw the angles
at the given points.

a) ∠ABC = 75° b) ∠EFG = 45° E


AB F

c) ∠PQR = 120° d) ∠XYZ = 135° X
Q

P
Y

4. a) Let's draw a line segment AB = 5 cm. Construct ∠BAC = ∠ABC = 60°.
Now, draw the perpendicular bisectors of each side of ∆ ABC. Do all the
perpendicular bisectors intersect at the same point?

b) Let's draw a line segment XY = 5.5 cm. Construct 105° and 30° respectively
at the points X and Y. Name the point of intersection of the arms of these
angles as Z and measure the size of ∠XZY.

c) Let's draw a line segment PQ = 6.2 cm. Construct ∠QPR = 60° and
∠PQR = 45°. Now, draw the angular bisectors of each angle of ∆PQR. Do all
the angular bisectors intersect at the same point?

It's your time - Project Work!
5. Let's draw three horizontal line segments and construct three different pairs of

angles at two points of each line segment. Do the arms of two angles drawn on
each line segment intersect?

14.4 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 O B
common vertex O and a common arm OB. AO A
They are called adjacent angles. Thus, a
pair of angles having a common vertex and B
a common arm are called adjacent angels. A

(ii) Linear pair

In the given figure, ∠AOB and ∠BOC are a pair of adjacent 180°
angles. Their sum is a straight angle (180°),

i.e. ∠AOB + ∠BOC = 180° CO
∠AOB and ∠BOC are called a linear pair.

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

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(iii) Vertically opposite angles AC

In the given figure, ∠AOC and ∠BOD are formed by

intersected line segments and they lie to the opposite O B
side of the common vertex. They are called vertically D

opposite angles. ∠AOD and ∠BOC are another pair of

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
(90°), i.e., ∠AOB + ∠BOC = 90°

∠AOB and ∠BOC are called complementary angles.
Thus, a pair of angles whose sum is 90° are called complementary 90°

angles. The complementary angles may or may not be adjacent. O A
Here, complement of ∠AOB = 90° – ∠BOC.

complement of ∠BOC = 90° – ∠AOB

(v) Supplementary angles B
In the given figure, the sum of ∠AOB and ∠BOC is two

right angles (180°), 180°

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

Thus, a pair of angles whose sum is 180° are called

supplementary angles. The supplementary angles may or may not be adjacent.

Here, supplement of ∠AOB = 180° – ∠BOC

Supplement of ∠BOC = 180° – ∠AOB.

14.5 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.
D
AD D B

AO B O

O

CB C A fig (iii) C

fig (i) 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.

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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 (ii) fig (iii)

(ii) Let's measure ∠AOC and ∠BOC with a protractor and write the measurements

in the table.

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.

C
A

B

OA O C OB

C B A fig (iii)
fig (i) fig (ii) Vedanta Excel in Mathematics - Book 7

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Geometry: Lines and Angles

(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 2x° and (x + 30)° are a pair of complementary angles, find them.

Solution:

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

or, 3x° = 90° – 30°

or, x° = 60°
3

= 20°

∴ 2x° = 2 × 20° = 40° and (x + 30)° = (20° + 30°) = 50°

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

Solution:

Let the required supplementary angles be 7x° and 3x°.

Now,

7x° + 3x° = 180° [The sum of a pair of supplementary angles]

or, 10x° = 180°

or, x° = 180°
10

= 18°

∴ 7x°= 7 × 18° = 126° and 3x° = 3 × 18° = 54°

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

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

or, x° = 11850°= 12°

∴ 4x° = 4 × 12 = 48°, 5x° = 5 × 12° = 60° and 6x° = 6 × 12° = 72°.

Again, p° = 4x° = 48°, q = 5x° = 60° and r° = 6x° = 72° [Each pair is vertically opposite
angles]

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EXERCISE 14.2

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?
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.

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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)
x
(x+5)° x° 4x° 2x°x° z° x° 3a°

y° a°
(x+25)° 3x° y°
30°
m) n) 88° o) p)

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

z° x° 2b°


Creative Section - B

8. a) In the figure given alongside, if x = 1 y,
3

show that y = 135° y° x°

b) In the adjoining figure, ∠XOY = 2∠YOZ, and X
Y
∠WOX = 3∠YOZ. Show that ∠WOX = 90°. W
OZ

c) In the figure alongside, if a + b + c = 180°, a
show that x = a + b.
b x
c

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a

d) In the given figure, if ∠a = ∠b, prove that ∠x = ∠y. 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°.

14.4 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 P
ab
the parallel lines. They are called exterior angles. ∠a and
M
∠c are lying to opposite side of the transversal. They are A B
N D
called alternate exterior angles. Similarly ∠b and ∠d are dc
Q
another pair of alternate exterior angles. C

Thus, alternate exterior angles are the pair of non-adjacent

exterior angles which lie to the opposite side of transversal.

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

Geometry: Lines and Angles

The alternate exterior angles made by transversal with parallel lines are always
equal.

∴ ∠a = ∠c and ∠b = ∠d

(ii) Interior and alternate interior angles

In the given figure, ∠a, ∠b, ∠c, and ∠d are lying P
M
inside the parallel lines. They are called interior A ab B
angles. ∠a and ∠c are lying to the opposite side of the dc D

transversal. They are called alternate interior (or simply C N
alternate) angles. Similarly, ∠b and ∠d are another pair
Q
of alternate angles.

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

(iii) Corresponding angles A P BA P B
C ab DC M D
∠a is an exterior and ∠c is an cd
interior angles lying to the same M
side of the transversal and they N
are not adjacent to each other. cd ab
They are called corresponding N
angles. ∠b and ∠d are another Q
pair of corresponding angles. Q

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

(iv) Co-interior angles

In the figure given alongside, ∠a and ∠d are a pair of interior P

angles lying to same side of the transversal. They are called A M B
co-interior (consecutive interior) angles. Similarly, ∠b and ab D

∠c are another pair of co-interior angles. C dc
N
Thus, a pair of interior angles lying on the same
Q
side of transversal are said to be co-interior angles.
Co-interior angles always lie in the ' '-shaped of parallel lines.

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Geometry: Lines and Angles

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

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°
Solution: 100° y
z x
25°

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

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

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

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

or, 100° + z = 180°

or, z = 180° – 100° = 80°

So, w = x = y = 100° and z = 80°

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

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

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

or, ∠x = 360° − 65°

∴ ∠x = 295°

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

Geometry: Lines and Angles

EXERCISE 14.3
General Section - Classwork
1. From the given figure, let's say 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
d) In the adjoining figure, which of the following is not true?
(i) ∠m + ∠x =180° ab
(ii) ∠a + ∠y = 180° nm
(iii) ∠n + ∠p = 180° pq
(iv) ∠a + ∠q = 180° yx

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3. Each of the pair of these angles are formed between parallel lines. Let's say
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

Z YD

P QX Y W T A B
CQ R X
B

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

a) b) c) d) c°

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

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°


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Geometry: Lines and Angles

i) c° j) k) l) 55°

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


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


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

a) b) c) d)

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



Creative Section - B

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

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

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 = 360°. y° c° b°

d° a° x°

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

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Geometry: Lines and Angles

e) In the given figure, show that ∠a + ∠b = ∠c + ∠d b° a°
d°

c°

F E
A 100°

f) In the figure alongside, find ∠x and ∠y, and x° y°
show that BF and CE are parallel to each other. B 45° C 55°D

10. a) In the given figure, AB // DE, ∠BAC = 120° and A B
∠CDE = 150°, find the value of x° 120° E

[Hint: Produce ED to intersect AC at F.] FD X
150° Y
b) From the figure given alongside, find the value of x.
[Hint: Through C, draw CZ // DY // BX] C

A
B 3x°

C 90°
D 2x°
E

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

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Geometry: Lines and Angles

OBJECTIVE QUESTIONS

Let’s tick (√) the correct alternative.

1. Adjacent angles have (B) common vertex (C) both (A) and (B) (D) none
(A) a common arm

2. Linear pair are always

(A) adjacent (B) equal (C) unequal (D) right angles

3. If ∠x and ∠y are linear pair, what is the value of ∠x+∠y?

(A) 90o (B) 180o (C) 270o (D) 360o

4. Which of the following statements is true?

(A) The vertically opposite angles are always equal.

(B) The complementary angles are always adjacent.

(C) The supplementary angles are always adjacent.

(D) The adjacent angles are always complementary.

5. If 5x° and (x + 36°) are linear pair, then x° is

(A) 9o (B) 24o (C) 34o (D) 44o

6. If 2xo and 3xo are complementary angles, what is the measure of bigger angle?

(A) 36o (B) 54o (C) 72o (D) 108o

7. The supplement of 60° is

(A) 30o (B) 180o (C) 120o (D) 90o

8. Which of the following statements is true?

(A) The alternate angles are always equal.

(B) The corresponding angles are always equal.

(C) The co-interior angles between parallel lines are always complementary.

(D) The co-interior angles between parallel lines are always supplementary.

9. If (x+20o) and (3x – 80o) are alternate angles between parallel lines, what is the value
of x?

(A) 30o (B) 50o (C) 60o (D) 75o

10. If (p - 50o) and (90o – p) are corresponding angles between parallel lines, then the value
of p is

(A) 20o (B) 70o (C) 80o (D) 55o

Vedanta ICT Corner
Please! Scan this QR code or
browse the link given below:

https://www.geogebra.org/m/tuyagzhv

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Unit 15 Plane Figures

15.1 Triangles – Looking back
Classwork - 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. ...................................................

15.2 Construction of triangles
A triangle is a closed plane figures bounded by three line segments. It has three
sides, three vertices, and three angles. In the given 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 DABC.

Let’s learn to construct the various types of triangles under the following conditions.

1. When the lengths of 3 sides are given C

Construct a triangle ABC in which AB = 5.5 cm, 4.6 cm

BC = 4.6 cm and CA = 3.8 cm 3.8 cm

Steps of construction

(i) Draw a line segment AB = 5.5 cm. A 5.5 cm B

(ii) With centre at B and radius 4.6 cm, draw an arc.

(iii) With centre A and radius 3.8 cm, draw another arc to intersect the first arc
at C.

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

Thus, ABC is the required triangle.

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2. When the lengths of two sides and angle made by them are given

Construct a triangle ABC in which

AB = 5.2 cm, AC = 4.5 cm and ∠A = 45°. X
C
Steps of construction
(i) Draw a line segment AB = 5.2 cm 120° 90° 60°
(ii) At A, construct ∠BAX = 45°

(iii) With centre at A and radius 4.5 cm, draw 45° 5.2 cm B
an arc to cut AX at C. A

(iv) Join B and C.

Thus, ABC is the required triangle.

3. When two angles and their common adjacent side are given

Construct a triangle ABC in which AB = 4.8 cm, ∠A = 60°, and ∠B = 30°.

Steps of construction

(i) Draw a line segment AB = 4.8 cm

(ii) At A and B construct ∠BAX = 60° and ∠ABY = 30° respectively. AX and

BY intersect each other at C. YC X

Thus, ABC is the required triangle.

60° 30° B
A 4.8 cm

4. When lengths of hypotenuse and a side of a right angled triangle are given

Construct a right angled triangle ABC in which a side AB = 4.5 cm, and

hypotenuse AC = 6 cm. X
Steps of construction

(i) Draw a line segment AB = 4.5 cm C
(ii) At B, construct ∠ABX = 90° 6 cm
(iii) With centre A and radius 6 cm, draw an arc to

intersect BX at C. 90°
B
(iv) Join A and C. A 4.5 cm

Thus, ABC is the required right-angled triangle.

EXERCISE 15.1

1. Let's construct triangle ABC under the following conditions.

a) AB = 4 cm, BC = 4.5 cm, CA = 5 cm

b) BC = 5.5 cm, CA = 4.8 cm, AB = 4.2 cm

c) AC = 6.5 cm, AB = 5.4 cm, BC = 4.8 cm

2. Let's construct triangle PQR in which

a) PQ = 4.7 cm, ∠P = 60°, PR = 5.3 cm

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b) QR = 5.6 cm, ∠Q = 45°, RP = 6.2 cm

c) PR = 4.5 cm, ∠R = 75°, PQ = 5.8 cm

3. Let's construct triangle XYZ in which

a) XY = 3.8 cm, ∠X = 60°, ∠Y = 45°

b) YZ = 4.6 cm, ∠Y = 120°, ∠Z = 30°

c) XZ = 5.3 cm, ∠X = 75°, ∠Z = 45°

4. Let's construct the right angled triangle ABC in which

a) AB = 4.5 cm, hypotenuse AC = 5.7 cm

b) AB = 3 cm, hypotenuse BC = 5 cm
c) BC = 5 cm, hypotenuse AB = 6.5 cm

15.3 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.

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 Q P
The sum of the angles of a triangle is always 180°.
In ∆PQR, ∠P + ∠Q+ ∠R = 180° R
P
Property 4
R
In an isosceles triangle, the angles opposite to the equal sides are

always equal.

Q

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In the figure, sides PQ = PR, So, ∠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 2x°, 3x°, and 4x° respectively.

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

or, 9x° = 180°

or, x° = 180°
9

= 20°.

∴ 2x° = 2 × 20 = 40°, 3x° = 3 × 20° = 60°, and 4x° = 4 × 20° = 80°.

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, 5x° = 90°

or, x° = 90° = 18°
5

Now, 2x° = 2 × 18° = 36°, 3x° = 3 × 18° = 54°

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

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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 + 110° = 180° [Being the sum a straight angle]

or, x = 180° – 110° = 70°

y = x = 70° [Being the base angles of an isosceles triangle]

Again, y + z = 110° [Being the sum equal to exterior angle of the

triangle]

or, 70° + z = 110°

or, z = 110° – 70° = 40°

∴ x = y = 70° and z = 40°.

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

or, a = 180° – 132° = 48°

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

or, b + 25° = 48°

or, b = 48° – 25° = 23°

∴a = 48° and b = 23°.

EXERCISE 15.2
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. .............................

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

b 6.5 cm Q D

R

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

greatest angle is ......... greatest angle is .........
smallest angle is ......... smallest angle is .........

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dX

YZ M

longest side is ......... longest side is ......... AB
shortest side is ......... shortest side is .........
longest side is .........
shortest side is .........

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

60° 50°

x° 40° y° = .............. z°
z° = ..............
x° = ..............

4. Let's say 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.
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.

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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°

e) f) g) h)

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

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It's your time - Project work!

10. 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.

11. 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.

15.4 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. DC
∴ AB = DC and AB // DC, AD = BC and AD // BC
AB
Its opposite angles are equal. DC
∴ ∠A = ∠C and ∠B = ∠D O

Its diagonals bisect each other. AB

∴ Diagonals AC and BD bisect each other at O.

i.e. AO = OC and BO = OD.

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(ii) Rectangle Plane Figures
Its opposite sides are equal and parallel.
DC
∴ AB = DC and AB // DC, AD = BC, and AD // BC.
Its all angles are equal and each of them is 90°. AB
DC
∠A = ∠B = ∠C = ∠D = 90°
Its diagonals are equal and they bisect each other. O
CB
∴ Diagonals AC = BD, AO = OC, and BO = OD.

(iii) Square

Its all sides are equal D C
B
∴ AB = BC = CD = DA A 90° C
Its all angles are equal and they are 90° D B

∴ ∠A = ∠B = ∠C = ∠D = 90° C

Its diagonals are equal and they bisect each other at right angle. A O C

∴ AC = BD and they bisect each other at O at right angle.

i.e., AO = OC, BO = OD, and BD ⊥ AC at O.

(iv) Rhombus B

Its all sides are equal and opposite sides are parallel. A
∴ AB = BC = CD = DA and AB // DC, AD // BC
D
Its opposite angles are equal. B

∴ ∠A = ∠C and ∠B = ∠D A O
D
Its diagonals are not equal but they bisect each other at right
angle.

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

i.e., AO = OC, BO = OD and BD ⊥ AC at O.

(v) Trapezium DC
Its any one pair of opposite sides are parallel.
AB
∴ AB // DC. B
(vi) Kite
AC
Its particular pairs of adjacent sides are equal. D
AB = AD and BC = DC

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The opposite angles formed by each pair of B C
unequal adjacent sides are equal.
∠ABC = ∠ADC. AO
D
Diagonals intersect each other at right angle.
AC and BD intersect at O at right angle.
i.e. BD ⊥ AC at O.

15.5 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.

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 Fig (i) Q SR P
Fig (ii) R

Fig (iii) S

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(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 OO

A Fig (i) B B C 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.
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

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(ii) Write the measurements in the table.

Fig. Result
Length of diagonals

No.

PR QS
(i) PR = QS
(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.

DC CB A
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 ÐAOB Result
AO OC BO OD

(i) AO = OC = BO = OD
(ii)
(iii) ÐAOB = ÐBOC = ÐCOD = ÐDOA = 90°

Conclusion: The diagonals of square bisect each other perpendicularly.

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

SR PS P
Q

OO
OS

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

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(ii) Write the measurements in the table.

Fig. Result
Parts of ÐQPS Parts of ÐPQR Parts of ÐQRS Parts of ÐPSR

No.

ÐQPO ÐSPO ÐPQO ÐRQO ÐQRO ÐSRO ÐRSO ÐPSO ÐQPO = ÐSPO
(i) ÐPQO = ÐRQO
ÐQRO = ÐSRO
(ii) ÐRSO = ÐPSO

(iii)

Conclusion: The diagonals of square bisect the vertical angles.

Worked-out examples

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

a) D (x+1)cm C b) S R c) A (2x−1)cm (x+3)cm D
120°

(2x+30)° B C

A (3x−5)cm B PQ

Solution:

a) Here, AB = CD [ Being opposite sides of parallelogram ]

or, 3x – 5 = x + 1

or, 3x – x = 1 + 5 Vedanta ICT Corner
or, Please! Scan this QR code or
2x = 6 browse the link given below:
or,
∴ x = 6 = 3 https://www.geogebra.org/m/pudrdujp
2

x =3

b) Here, ∠ P = ∠ R [Being opposite angles of parallelogram ]
or,
or, (2x + 30)° = 120°
or,
2x° = 120° – 30°
or,
2x° = 90°
∴ 90°
x° = 2 = 45°

x° = 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

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Example 2: If x°, (x + 20)°, (x + 40)°, and (x + 60)° are the angles of a quadrilateral,
find them.

Solution:

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

or, (4x + 120)° = 360°

or, 4x° = 360° – 120°

or, x = 240° = 60°
4

∴ x° = 60° , (x + 20)° = 60° + 20° = 80° , (x + 40)° = 60° + 40° = 100°

and (x + 60)° = 60° + 60° = 120°

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 x°, 2x°, 3x°, and 4x° respectively.

Here, x° + 2x° + 3x° + 4x° = 360°

or, 10x° = 360°

or, x° = 360° = 36°
10
∴ x ° = 36°, 2x° = 2 × 36° = 72°, 3x° = 3 × 36° = 108° and 4x° = 4 × 36° = 144°.

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:

a) y° + 105° = 180° [Being the sum a straight angle]

or, y° = 180° – 105° = 75°

Now, x° + (2x° + 10°) + 2x° + y° = 360° [The sum of the angles of a quadrilateral]

or, 5x° + 10° + 75° = 360°

or, 5x° = 360° – 85°

or, x° = 275° = 55°
5

∴ x° = 55°, 2x° + 10° = 2 × 55° + 10° = 120°, 2x° = 2 × 55° = 110° and y° = 75°.

b) Here, w° = 25° [Being alternate angles]

x° = 20° [Being alternate angles]

Now, w° + 20° = 25° + 20° = 45°

x° + 25° = 20° + 25° = 45°

Again, y° + 45° = 180° [Being the sum of co-interior angles]
[Being the opposite angles of a parallelogram]
or, y° = 180° – 45°

or, y° = 135°

Also, z° = y° = 135°

∴w° = 25°, x° = 20°, y° = z° = 135°.

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EXERCISE 15.3
General Section - Classwork
1. Let's say 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 say 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° = .....................

(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 say and write the unknown angles as quickly as possible.

a) b° a° b) c) 65° x°

55° x°

a° = .......... x° x° = .......... 60° y°
b° = ..........
x° = ..........
y° = ..........

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

e) f) g) h) W
V
D 5 cm C PO S RX

(x – 3) cm (x+4)cm 7cm QU (3x–5()xc+m5)cm
(7– x) cm P

A (2x – 3) cm B M N

6. Let's work out these problems.

a) If a°, 60°, 120°, and 130° are the angles of a quadrilateral, find a°.

b) If x°, 2x°, 85,° and 140° are the angles of a quadrilateral, find x° and 2x°.

c) If a°, 2a°, 4a°, and 5a° are the angles of a quadrilateral, find them.

d) If x°, 3x°, (x° + 15°), and (2x° + 30°) 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°

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e) y° f) b° c° g) 100° h) x°

z° 110° a° 4c° y° x° 130°
z° z°+10° 2x°
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°


It's your time - Project work
9. 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.

15.6 Pythagoras Theorem A

The adjoining triangle is a right angled triangle ABC right angled at B. Perpendicular Hypotenuse
The side AC opposite to the right angle is called its hypotenuse. The
sides AB and BC are called its perpendicular and base respectively.

Pythagoras was a Greek Mathematician who lived around 2500 years Base C
ago. He discovered a significant fact about right-angled triangles B
known as Pythagoras Theorem.

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Pythagoras Theorem states that in a right angled triangle, the square of the hypotenuse
is equal to the sum of the squares of perpendicular and base.

∴ (hypotenuse)2 = (perpendicular)2 + (base)2 F
h2 = p2 + b2 A

Study the following illustration and try to understand the G I
Pythagoras Theorem. B

In the given figure, DABC is a right angled triangle, H
right angled at B. Here AC = 5 cm is the hypotenuse,
AB = 4 cm is perpendicular and BC= 3 cm is base. C

Here, area of the square ACDE = 25 sq. cm.

Area of square BCHI = 9 sq. cm. ED
Area of square ABFG = 16 sq. cm.

Thus, area of square ACDE = area of square BCHI + area of square ABFG

i.e. square of hypotenuse = square of perpendicular + square of base

h2 = p2 + b2.
Experiment: Experimental verification of Pythagoras Theorem.

Step 1: Draw three right angled triangles ABC of different sizes and right angled at B.

A
B BA

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

Step 2: Measure the lengths of the sides of each triangle and write the measurements in

the table.

Fig No. AB BC CA AB2 BC2 CA2 AB2 + BC2 Results

(i) CA2 = AB2 + BC2

(ii) CA2 = AB2 + BC2

(iii) CA2 = AB2 + BC2

Conclusion: The square of the hypotenuse of a right-angled triangle is equal to the sum
of the squares of perpendicular and base.

Worked-out examples

Example 1: In the adjoining right angled triangle ABC, calculate the length of AC.

Solution A

Here, base (b) = BC = 8 cm 6 cm

perpendicular (p) = AB = 6 cm

hypotenuse (h) = AC = ? B 8 cm C

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Now, by using the Pythagoras theorem,
h2 = p2 + b2
or, h2 = 62 + 82
or, h2 = 36 + 64
or, h2 = 100
or, h = 100 = 10 cm
∴ The required length of AC is 10 cm.

Example 2: In the given figure, a ladder 5 m long rest against a vertical wall. If the
height of the wall at which the upper end of the ladder is supported is
3 m, find the distance of the foot of the ladder from the wall.

Solution:
Let AC be the length of the ladder, AB be the height of the wall, and BC be the distance
between the wall and foot of the ladder.

Here, hypotenuse (h) = AC = 5 m

perpendicular (p) = AB = 3 m A

base (b) = BC = ?

Now, by using Pythagoras theorem, 5m 3m
h2 = p2 + b2

or, 52 = 32 + b2

or, 25 = 9 + b2 CB

or, b2 = 25 – 9 = 16
or, b = 16 = 4 m.

So, the required distance is 4 m.

EXERCISE 15.4
General Section – Classwork

1. Let's say and write the 'True' or 'False' for the following statements.

a) Every right-angled triangle has perpendicular, base and hypotenuse. ...........

b) h2 = p2 + b2 is valid for every triangle. ........................

c) Pythagoras theorem is valid only for every right-angled triangle. ...................

d) Hypotenuse is the longest side in every triangle. ........................

e) Hypotenuse is the opposite side of right angle of a right-angled triangle.
........................

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f) If the sides of a triangle are 3 cm, 4 cm and 6 cm, the triangle is a right-
angled triangle. ........................

g) If the sides of a triangle are 3 cm, 4 cm and 5 cm, the triangle is a right-
angled triangle. ........................

2. Let's say and write the answers as quickly as possible. In any right-angled
triangle if:

a) p = 3, b = 4, h = ............ b) h = 5, p = 3, b = ............

c) h = 5, b = 4, p = ............ d) p = 8, b = 6, h = ............
Creative Section - A

3. Find the unknown lengths of sides of the following right angled triangles.

a) b) c) 12 cm d) M
A RX K
Q
12 cm
4 cm ZL
6 cm 10 cm 15 cm
9 cm
B 3 cm P Y
C 13 cm

4. Using Pythagoras theorem, find whether the following sides of triangles are the
sides of right angled triangles.

a) 3 cm, 4 cm, 5 cm b) 6 cm, 8 cm, 10 cm c) 9 cm, 12 cm, 16 cm

d) 7 cm, 10 cm, 14 cm e) 5 cm, 5 3 cm, 10 cm f) 3 3 cm, 6 cm, 8 cm

B

5. a) In the adjoining figure, ABC is a triangle. If 15 cm 9 cm
BD ⊥ AC, AB = BC = 15 cm, BD = 9 cm, find the D
length of AC. A C

b) Find the length of diagonal of the adjoining rectangle. 12 cm

16 cm

c) If the length and breadth of a rectangle are 12 cm and 5 cm respectively, what
will be the length of its diagonal?

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Creative Section -B S 13 cm
x cm
6. a) In the adjoining figure, find the values of x and y. y cm R
3 cm

P 4 cm Q

B

b) Find the length of x and y in the adjoining A 13 cm x cm E 5 3cm C
diagram. 12 cm

D y cm
A

7. a) A ladder 10 m long rests against a vertical wall 6 m 10 m 6m

above the ground. At what distance does the ladder

touch the ground from the bottom of the wall?

B C
C
A

b) The vertical height of a tree is 18 m. The uppermost part

of the tree is broken by wind 5 m above the ground. At

what distance does the uppermost part of the tree touch

the ground from the foot of the tree? B

c) In the figure alongside, ABCD is a rectangular ground B C

of length 40 m and breadth 30 m. Ramesh reached at 30 m
C walking from A to B and B to C. But Shyam reached

to C walking from A to C directly. Who walked the A 40 m D
shorter distance and by how much?

d) In the figure alongside, AB is an electric pole. When an A

electric wire falls, its one end touches the ground at P,

5 m away from the base of the pole. If the length of the wire 13 m

from the top of the pole to the ground is 13 m, complete the

following problems. P 5m B 9m Q
(i) Find the height of the pole.

(ii) If the other end of the wire touches the ground at Q, 9 m away from the base
to the opposite side of the pole, find the length of the falling wire.

Vedanta ICT Corner
Please! Scan this QR code or
browse the link given below:

https://www.geogebra.org/m/v4y6ufpk

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It's your time - Project work!

8. a) Let's fold a rectangular sheet of paper through its diagonal and cut out a right-

angled triangle. Measure the length of perpendicular, base and hypotenuse of

the triangle by using a ruler and verify that h2 = p2 + b2.

b) Let's fold a square sheet of paper through its diagonal and cut out a triangle.
Measure the lengths of all 3 sides of the triangle by using a ruler. Apply the
Pythagoras theorem and verify whether the triangle so formed is a right-angled
triangle or not.

OBJECTIVE QUESTIONS

1. The sum of interior angles of a triangle is always

(A) Less than 180o (B) more than 180o (C) equal to 180o (D) 1 right angle

2. If two angles of a triangle are 50o and 60o, what is the third angle?

(A) 40o (B) 60o (C) 70o (D) 100o

3. Which of the following is not the property of the parallelogram?

(A) Its opposite sides are equal and parallel. (B) Its opposite angles are equal.

(C) Its diagonals bisect to each other. (d) Its diagonals are equal.

4. Which of the following is the property of the rectangle?

(A) Its opposite sides are equal and parallel. (B) It’s all angles are equal and 90o.

(C) Its diagonals are equal. (D) All of the above.

5. The sum of angles of a quadrilateral is

(A) 90o (B) 180o (C) 270o (D) 360o

6. Which of the following is the property of the square?

(A) It’s all sides and angles are equal. (B) Its diagonals are equal.

(C) Its diagonals are perpendicular bisector to each other. (D) All of the above.

7. Which of the following statements is true?

(A) Every parallelogram is a rectangle. (B) Every rectangle is a rhombus.

(C) Every rhombus is a square. (D) Every square is a rhombus.

8. If p, b and h are the perpendicular, base and hypotenuse of a right angled triangle
respectively, then which of the following is the Pythagoras theorem?

(A) p + b = h (B) p2 + b2 = h2 (C) p2 + h2 = b2 (D) b2 + h2 = p2

9. If (x + 10) cm and (2x + 3) cm are the opposite sides of parallelogram, then the value
of x is

(A) 7 cm (B) 13 cm (C) 17 cm (D) 20 cm

10. If (3y + 8) cm and (y + 12) cm are the length of diagonals of a rectangle, then the value
of y is

(A) 2 cm (B) 14 cm (C) 8 cm (D) 10 cm

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