RS Aggarwal Solutions: Angles and Their Measurement | Mathematics (Maths) Class 6 PDF Download

 Page 1


 
Points to Remember :
Angle. The figure formed by two rays with the same initial point,
is called the angle. The common initial point is called the vertex
of the angle and the rays forming the angle are called its arms or
sides. In the figure, rays OA and OB with common initial point
O form the angle AOB or angle BOA. We denote it as AOB or
BOA or simply O.
Interior And Exterior of An Angle
Let AOB be an angle. Then, the set of all the points which lie
inside the angle, form the interior of the angle. Any point on any
of the arms of the angle is said to lie on the angle. The set of all
those points which lie outside the angle, form the exterior of the
angle.
In the figure, P lies in the interior of AOB, Q lies on AOB and R lies in the exterior of AOB.
Sol. (i) In the given figure, three angles are
formed.
Q. 1. Name three examples of angles from
your daily life.
Sol. Three examples are : Tongs, Scissors
and Compasses.
Q. 2. Name the vertex and the arms of ABC,
given in the figure below :
Sol. In the given angle ABC, the vertex is B
and arms are BA and BC .
Q. 3. How many angles are formed in the figure
given below ? Name them.
Page 2


 
Points to Remember :
Angle. The figure formed by two rays with the same initial point,
is called the angle. The common initial point is called the vertex
of the angle and the rays forming the angle are called its arms or
sides. In the figure, rays OA and OB with common initial point
O form the angle AOB or angle BOA. We denote it as AOB or
BOA or simply O.
Interior And Exterior of An Angle
Let AOB be an angle. Then, the set of all the points which lie
inside the angle, form the interior of the angle. Any point on any
of the arms of the angle is said to lie on the angle. The set of all
those points which lie outside the angle, form the exterior of the
angle.
In the figure, P lies in the interior of AOB, Q lies on AOB and R lies in the exterior of AOB.
Sol. (i) In the given figure, three angles are
formed.
Q. 1. Name three examples of angles from
your daily life.
Sol. Three examples are : Tongs, Scissors
and Compasses.
Q. 2. Name the vertex and the arms of ABC,
given in the figure below :
Sol. In the given angle ABC, the vertex is B
and arms are BA and BC .
Q. 3. How many angles are formed in the figure
given below ? Name them.
Names of the angles are :
ABC, BAC and ACB
(ii) In the given figure, four angles are
formed.
Names of the angles are :
ABC, BCD, CDA and BAD
(iii) In the given figure, eight angles are
formed.
Names of the angles are :
ABC, BCD, CDA, BAD, ABD,
DCB, ADBand BDC
Q. 4. In Fig. list the points which :
(i) are in the interior of AOB
(ii) are in the exterior of AOB
(iii) lie on AOB
Sol. In the given figure :
(i) Points S and Q are in the interior of
AOB
(ii) Points P and R are in the exterior of
AOB.
(iii) Points A, O, B, N, T lie on AOB.
Q. 5. See the adjacent Fig. and state which of
the following statements are true and
which are false
(i) Point C is in the interior of AOC
(ii) Point C is in the interor of AOD
(iii) Point D is in the interior of AOC
(iv) Point B is in the exterior of AOD.
(v) Point C lies on AOB.
Sol. (i) False (ii) True
(iii) False (iv) True
(v) False
Q. 6. In the adjoining figure write another
name for :
(i) 1 (ii) 2
(iii) 3
Sol. In the given figure, another name for :
(i) 1 is EPB (ii) 2 is PQC
Page 3


 
Points to Remember :
Angle. The figure formed by two rays with the same initial point,
is called the angle. The common initial point is called the vertex
of the angle and the rays forming the angle are called its arms or
sides. In the figure, rays OA and OB with common initial point
O form the angle AOB or angle BOA. We denote it as AOB or
BOA or simply O.
Interior And Exterior of An Angle
Let AOB be an angle. Then, the set of all the points which lie
inside the angle, form the interior of the angle. Any point on any
of the arms of the angle is said to lie on the angle. The set of all
those points which lie outside the angle, form the exterior of the
angle.
In the figure, P lies in the interior of AOB, Q lies on AOB and R lies in the exterior of AOB.
Sol. (i) In the given figure, three angles are
formed.
Q. 1. Name three examples of angles from
your daily life.
Sol. Three examples are : Tongs, Scissors
and Compasses.
Q. 2. Name the vertex and the arms of ABC,
given in the figure below :
Sol. In the given angle ABC, the vertex is B
and arms are BA and BC .
Q. 3. How many angles are formed in the figure
given below ? Name them.
Names of the angles are :
ABC, BAC and ACB
(ii) In the given figure, four angles are
formed.
Names of the angles are :
ABC, BCD, CDA and BAD
(iii) In the given figure, eight angles are
formed.
Names of the angles are :
ABC, BCD, CDA, BAD, ABD,
DCB, ADBand BDC
Q. 4. In Fig. list the points which :
(i) are in the interior of AOB
(ii) are in the exterior of AOB
(iii) lie on AOB
Sol. In the given figure :
(i) Points S and Q are in the interior of
AOB
(ii) Points P and R are in the exterior of
AOB.
(iii) Points A, O, B, N, T lie on AOB.
Q. 5. See the adjacent Fig. and state which of
the following statements are true and
which are false
(i) Point C is in the interior of AOC
(ii) Point C is in the interor of AOD
(iii) Point D is in the interior of AOC
(iv) Point B is in the exterior of AOD.
(v) Point C lies on AOB.
Sol. (i) False (ii) True
(iii) False (iv) True
(v) False
Q. 6. In the adjoining figure write another
name for :
(i) 1 (ii) 2
(iii) 3
Sol. In the given figure, another name for :
(i) 1 is EPB (ii) 2 is PQC
(iii) 3 is FQD
EXERCISE 13B
Q. 1. State the type of each of the following
angles :
Sol. (i) Obtuse angle  (ii) Right angle
(iii) Straight line (iv) Reflex angle
(v) Acute angle (vi) Complete angle
Q. 2. Classify the angles whose magnitudes are
given below :
(i) 30º (ii) 91º
(iii) 179º (iv) 90º
(v) 181º (vi) 360º
(vii) 128º (viii) (90·5)º
(ix) (39·3)º (x) 80º
(xi) 0º (xii) 15º
Sol. We know that an acute angle is less than
90º (ii) a right angle is equal to 90º (iii) an
obtuse angle is greater than 90º but less
than 180º (iv) an angle equal to 180º is a
straight angle (v) angle greater than 180º
but less than 360º is called a reflex angle
(vi) angle equal to 360º is called a
complete angle and angle equal to 0º is
called a zero angle. Now the angles are :
(i) acute (ii) obtuse
(iii) obtuse (iv) right
(v) reflex (vi) complete
(vii) obtuse (viii) obtuse
(ix) acute (x) acute
(xi) zero (xii) acute     Ans.
Q. 3. How many degrees are there in :
(i) One right angle ?
(ii) Two right anlges ?
(iii) Three right angles ?
(iv) Four right angles ?
(v)
2
3
 right angle ?
(vi) 1½ right angle ?
Sol. (i) One right angle = 90
(ii) Two right angles = (2 × 90) = 180
(iii) Three right angles = (3 × 90) = 270
(iv) Four right angles = (4 × 90) = 360
(v)
2
3
 right angle 
F
H
G
I
K
J
2
3
90 60
(vi) 1½ right angle 
F
H
G
I
K
J 1
1
2
90
 
F
H
G
I
K
J
3
2
90 135
Q. 4. How many degrees are there in the angle
between the hour hand and the minute
hand of a clock, when it is
(i) 3 o clock (ii) 6 o clock
(iii) 12 o clock (iv) 9 o clock
Sol. (i) When it is 3 o clock, the minute
hand is at 12, and hour hand is at 3 as
O B
A
C
O D
F O E
O
Q
P
O G
H
O
P
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Page 4


 
Points to Remember :
Angle. The figure formed by two rays with the same initial point,
is called the angle. The common initial point is called the vertex
of the angle and the rays forming the angle are called its arms or
sides. In the figure, rays OA and OB with common initial point
O form the angle AOB or angle BOA. We denote it as AOB or
BOA or simply O.
Interior And Exterior of An Angle
Let AOB be an angle. Then, the set of all the points which lie
inside the angle, form the interior of the angle. Any point on any
of the arms of the angle is said to lie on the angle. The set of all
those points which lie outside the angle, form the exterior of the
angle.
In the figure, P lies in the interior of AOB, Q lies on AOB and R lies in the exterior of AOB.
Sol. (i) In the given figure, three angles are
formed.
Q. 1. Name three examples of angles from
your daily life.
Sol. Three examples are : Tongs, Scissors
and Compasses.
Q. 2. Name the vertex and the arms of ABC,
given in the figure below :
Sol. In the given angle ABC, the vertex is B
and arms are BA and BC .
Q. 3. How many angles are formed in the figure
given below ? Name them.
Names of the angles are :
ABC, BAC and ACB
(ii) In the given figure, four angles are
formed.
Names of the angles are :
ABC, BCD, CDA and BAD
(iii) In the given figure, eight angles are
formed.
Names of the angles are :
ABC, BCD, CDA, BAD, ABD,
DCB, ADBand BDC
Q. 4. In Fig. list the points which :
(i) are in the interior of AOB
(ii) are in the exterior of AOB
(iii) lie on AOB
Sol. In the given figure :
(i) Points S and Q are in the interior of
AOB
(ii) Points P and R are in the exterior of
AOB.
(iii) Points A, O, B, N, T lie on AOB.
Q. 5. See the adjacent Fig. and state which of
the following statements are true and
which are false
(i) Point C is in the interior of AOC
(ii) Point C is in the interor of AOD
(iii) Point D is in the interior of AOC
(iv) Point B is in the exterior of AOD.
(v) Point C lies on AOB.
Sol. (i) False (ii) True
(iii) False (iv) True
(v) False
Q. 6. In the adjoining figure write another
name for :
(i) 1 (ii) 2
(iii) 3
Sol. In the given figure, another name for :
(i) 1 is EPB (ii) 2 is PQC
(iii) 3 is FQD
EXERCISE 13B
Q. 1. State the type of each of the following
angles :
Sol. (i) Obtuse angle  (ii) Right angle
(iii) Straight line (iv) Reflex angle
(v) Acute angle (vi) Complete angle
Q. 2. Classify the angles whose magnitudes are
given below :
(i) 30º (ii) 91º
(iii) 179º (iv) 90º
(v) 181º (vi) 360º
(vii) 128º (viii) (90·5)º
(ix) (39·3)º (x) 80º
(xi) 0º (xii) 15º
Sol. We know that an acute angle is less than
90º (ii) a right angle is equal to 90º (iii) an
obtuse angle is greater than 90º but less
than 180º (iv) an angle equal to 180º is a
straight angle (v) angle greater than 180º
but less than 360º is called a reflex angle
(vi) angle equal to 360º is called a
complete angle and angle equal to 0º is
called a zero angle. Now the angles are :
(i) acute (ii) obtuse
(iii) obtuse (iv) right
(v) reflex (vi) complete
(vii) obtuse (viii) obtuse
(ix) acute (x) acute
(xi) zero (xii) acute     Ans.
Q. 3. How many degrees are there in :
(i) One right angle ?
(ii) Two right anlges ?
(iii) Three right angles ?
(iv) Four right angles ?
(v)
2
3
 right angle ?
(vi) 1½ right angle ?
Sol. (i) One right angle = 90
(ii) Two right angles = (2 × 90) = 180
(iii) Three right angles = (3 × 90) = 270
(iv) Four right angles = (4 × 90) = 360
(v)
2
3
 right angle 
F
H
G
I
K
J
2
3
90 60
(vi) 1½ right angle 
F
H
G
I
K
J 1
1
2
90
 
F
H
G
I
K
J
3
2
90 135
Q. 4. How many degrees are there in the angle
between the hour hand and the minute
hand of a clock, when it is
(i) 3 o clock (ii) 6 o clock
(iii) 12 o clock (iv) 9 o clock
Sol. (i) When it is 3 o clock, the minute
hand is at 12, and hour hand is at 3 as
O B
A
C
O D
F O E
O
Q
P
O G
H
O
P
(i)
(ii)
(iii)
(iv)
(v)
(vi)
shown in the figure, clearly, the angle
between the two hands = 90 .
(ii) When it is 6 o clock, the minute hand is
at 12 and the hour hand is at 6 as shown
in the figure. Clearly, the angle between
the two hands of the clock is a straight
angle is i.e. 180º.
(iii) When it is 12 o clock, both the hands
of the clock lie at 12 as shown in the
figure. Clearly, the angle between the
two hands = 0 .
(iv) When it is 9 o  clock, the minute hand
is at 12 and the hour hand is at 9 as
shown in the figure. Clearly, the angle
between the two hands = 90º.
Q. 5. Using a rular, draw an acute angle, an
obtuse angle and a straight angle.
Sol. (i) Take the rular and draw any ray OA.
Again using the rular, starting from O,
draw a ray OB in such a way that the
angle formed is less than 90 . Then,
AOB is the required acute angle.
(ii) Take the rular and draw any ray OA.
Now, starting from O, draw another ray
OB, with the help of the rular, such that
the angle formed is greater than a right
angle.
Then, AOB is the required obtuse
angle.
(iii) Take a rular and draw any ray OA. Now,
starting from O, draw ray OB in the
opposite direction of the ray OA. Then
AOB is the required straight angle.
Page 5


 
Points to Remember :
Angle. The figure formed by two rays with the same initial point,
is called the angle. The common initial point is called the vertex
of the angle and the rays forming the angle are called its arms or
sides. In the figure, rays OA and OB with common initial point
O form the angle AOB or angle BOA. We denote it as AOB or
BOA or simply O.
Interior And Exterior of An Angle
Let AOB be an angle. Then, the set of all the points which lie
inside the angle, form the interior of the angle. Any point on any
of the arms of the angle is said to lie on the angle. The set of all
those points which lie outside the angle, form the exterior of the
angle.
In the figure, P lies in the interior of AOB, Q lies on AOB and R lies in the exterior of AOB.
Sol. (i) In the given figure, three angles are
formed.
Q. 1. Name three examples of angles from
your daily life.
Sol. Three examples are : Tongs, Scissors
and Compasses.
Q. 2. Name the vertex and the arms of ABC,
given in the figure below :
Sol. In the given angle ABC, the vertex is B
and arms are BA and BC .
Q. 3. How many angles are formed in the figure
given below ? Name them.
Names of the angles are :
ABC, BAC and ACB
(ii) In the given figure, four angles are
formed.
Names of the angles are :
ABC, BCD, CDA and BAD
(iii) In the given figure, eight angles are
formed.
Names of the angles are :
ABC, BCD, CDA, BAD, ABD,
DCB, ADBand BDC
Q. 4. In Fig. list the points which :
(i) are in the interior of AOB
(ii) are in the exterior of AOB
(iii) lie on AOB
Sol. In the given figure :
(i) Points S and Q are in the interior of
AOB
(ii) Points P and R are in the exterior of
AOB.
(iii) Points A, O, B, N, T lie on AOB.
Q. 5. See the adjacent Fig. and state which of
the following statements are true and
which are false
(i) Point C is in the interior of AOC
(ii) Point C is in the interor of AOD
(iii) Point D is in the interior of AOC
(iv) Point B is in the exterior of AOD.
(v) Point C lies on AOB.
Sol. (i) False (ii) True
(iii) False (iv) True
(v) False
Q. 6. In the adjoining figure write another
name for :
(i) 1 (ii) 2
(iii) 3
Sol. In the given figure, another name for :
(i) 1 is EPB (ii) 2 is PQC
(iii) 3 is FQD
EXERCISE 13B
Q. 1. State the type of each of the following
angles :
Sol. (i) Obtuse angle  (ii) Right angle
(iii) Straight line (iv) Reflex angle
(v) Acute angle (vi) Complete angle
Q. 2. Classify the angles whose magnitudes are
given below :
(i) 30º (ii) 91º
(iii) 179º (iv) 90º
(v) 181º (vi) 360º
(vii) 128º (viii) (90·5)º
(ix) (39·3)º (x) 80º
(xi) 0º (xii) 15º
Sol. We know that an acute angle is less than
90º (ii) a right angle is equal to 90º (iii) an
obtuse angle is greater than 90º but less
than 180º (iv) an angle equal to 180º is a
straight angle (v) angle greater than 180º
but less than 360º is called a reflex angle
(vi) angle equal to 360º is called a
complete angle and angle equal to 0º is
called a zero angle. Now the angles are :
(i) acute (ii) obtuse
(iii) obtuse (iv) right
(v) reflex (vi) complete
(vii) obtuse (viii) obtuse
(ix) acute (x) acute
(xi) zero (xii) acute     Ans.
Q. 3. How many degrees are there in :
(i) One right angle ?
(ii) Two right anlges ?
(iii) Three right angles ?
(iv) Four right angles ?
(v)
2
3
 right angle ?
(vi) 1½ right angle ?
Sol. (i) One right angle = 90
(ii) Two right angles = (2 × 90) = 180
(iii) Three right angles = (3 × 90) = 270
(iv) Four right angles = (4 × 90) = 360
(v)
2
3
 right angle 
F
H
G
I
K
J
2
3
90 60
(vi) 1½ right angle 
F
H
G
I
K
J 1
1
2
90
 
F
H
G
I
K
J
3
2
90 135
Q. 4. How many degrees are there in the angle
between the hour hand and the minute
hand of a clock, when it is
(i) 3 o clock (ii) 6 o clock
(iii) 12 o clock (iv) 9 o clock
Sol. (i) When it is 3 o clock, the minute
hand is at 12, and hour hand is at 3 as
O B
A
C
O D
F O E
O
Q
P
O G
H
O
P
(i)
(ii)
(iii)
(iv)
(v)
(vi)
shown in the figure, clearly, the angle
between the two hands = 90 .
(ii) When it is 6 o clock, the minute hand is
at 12 and the hour hand is at 6 as shown
in the figure. Clearly, the angle between
the two hands of the clock is a straight
angle is i.e. 180º.
(iii) When it is 12 o clock, both the hands
of the clock lie at 12 as shown in the
figure. Clearly, the angle between the
two hands = 0 .
(iv) When it is 9 o  clock, the minute hand
is at 12 and the hour hand is at 9 as
shown in the figure. Clearly, the angle
between the two hands = 90º.
Q. 5. Using a rular, draw an acute angle, an
obtuse angle and a straight angle.
Sol. (i) Take the rular and draw any ray OA.
Again using the rular, starting from O,
draw a ray OB in such a way that the
angle formed is less than 90 . Then,
AOB is the required acute angle.
(ii) Take the rular and draw any ray OA.
Now, starting from O, draw another ray
OB, with the help of the rular, such that
the angle formed is greater than a right
angle.
Then, AOB is the required obtuse
angle.
(iii) Take a rular and draw any ray OA. Now,
starting from O, draw ray OB in the
opposite direction of the ray OA. Then
AOB is the required straight angle.
EXERCISE 13 C
Q. 1. Measure each of the following angles
with the help of a protractor and write
the measures in degrees.
Sol. (i) Place the protractor in such a way
that its centre is exactly at the vertex O
of the given angle AOB and the base line
lies along the arm OA. Read off the mark
through which the arm OB passes,
starting from 0 on the side A.
We find that AOB = 45 .
(ii) The given angle is PQR. Place the
protractor in such a way that its centre
is exactly on the vertex Q of the given
angle and the base line lies along the arm
QR.Read  off the mark through which
the arm QP passes, starting from 0 on
the side of R.
We find that PQR = 67 .
(iii) The given angle is DEF. Place the
protractor in such  a way that its centre
is exactly on the vertex E of the given
angle and the base line lies along the arm
ED. Read off the mark through which
the arm EF passes, starting from 0 on
the side of D.
We find that the DEF = 130°.
(iv) The given angle is LMN. Place the
protractor in such a way that its centre
is exactly on the vertex M of the given
angle and the base line lies along the arm
ML. Read off the mark through which
the arm MN passes, starting from 0 on
the side of L.
We find that the LMN = 50°.
(v) The given angle is RST. Place the
protractor in such a way that its centre
is exactly on the vertex S of the given
angle and the base line lies along the arm
SR. Read off the mark through which
the arm ST passes, starting from 0° on
the side of R.