NCERT Textbook: Lines & Angles | NCERT Textbooks (Class 6 to Class 12) - CTET & State TET PDF Download

 Page 1


MATHEMATICS 74
Lines and
Angles
5.1 INTRODUCTION
You already know how to identify different lines, line segments and angles in a given
shape. Can you identify the different line segments and angles formed in the following
figures? (Fig 5.1)
(i) (ii) (iii) (iv)
Fig 5.1
Can you also identify whether the angles made are acute or obtuse or right?
Recall that a line segment has two end points. If we extend the two end points in either
direction endlessly, we get a line. Thus, we can say that a line has no end points. On the other
hand, recall that a ray has one end point (namely its starting point). For example, look at the
figures given below:
Fig 5.2
Here, Fig 5.2 (i) shows a line segment, Fig 5.2 (ii) shows a line and Fig 5.2 (iii) is that
of a ray. A line segment PQ is generally denoted by the symbol 
PQ
, a line AB is denoted by
the symbol AB

 
and the ray OP is denoted by OP
ur uu
. Give some examples of line segments and
rays from your daily life and discuss them with your friends.
(iii)
(i)
(ii)
Chapter  5
2024-25
Page 2


MATHEMATICS 74
Lines and
Angles
5.1 INTRODUCTION
You already know how to identify different lines, line segments and angles in a given
shape. Can you identify the different line segments and angles formed in the following
figures? (Fig 5.1)
(i) (ii) (iii) (iv)
Fig 5.1
Can you also identify whether the angles made are acute or obtuse or right?
Recall that a line segment has two end points. If we extend the two end points in either
direction endlessly, we get a line. Thus, we can say that a line has no end points. On the other
hand, recall that a ray has one end point (namely its starting point). For example, look at the
figures given below:
Fig 5.2
Here, Fig 5.2 (i) shows a line segment, Fig 5.2 (ii) shows a line and Fig 5.2 (iii) is that
of a ray. A line segment PQ is generally denoted by the symbol 
PQ
, a line AB is denoted by
the symbol AB

 
and the ray OP is denoted by OP
ur uu
. Give some examples of line segments and
rays from your daily life and discuss them with your friends.
(iii)
(i)
(ii)
Chapter  5
2024-25
LINES AND ANGLES 75
Again recall that an angle is formed when lines or line segments meet. In Fig 5.1,
observe the corners. These corners are formed when two lines or line segments intersect
at a point. For example, look at the figures given below:
Fig 5.3
In Fig 5.3 (i) line segments AB and BC intersect at B to form angle
ABC, and again line segments BC and AC intersect at C to form angle
ACB and so on. Whereas, in Fig 5.3 (ii) lines PQ and RS intersect at O
to form four angles POS, SOQ, QOR and ROP. An angle ABC is
represented by the symbol ?ABC. Thus, in Fig 5.3 (i), the three angles
formed are ?ABC, ?BCA and ?BAC, and in Fig 5.3 (ii), the four
angles formed are ? POS, ? SOQ, ? QOR and ? POR. Y ou have already
studied how to classify the angles as acute, obtuse or right angle.
Note: While referring to the measure of an angle ABC, we shall write m? ABC as simply
? ABC. The context will make it clear, whether we are referring to the angle or its
measure.
5.2 RELATED ANGLES
5.2.1  Complementary Angles
When the sum of the measures of two angles is 90°, the angles are called complementary
angles.
(i) (ii)
List ten figures around you
and identify the acute, obtuse
and right angles found in them.
TRY THESE
Whenever two angles are complementary , each angle is said to be the complement
of the other angle. In the above diagram (Fig 5.4), the ‘30° angle’ is the complement of the
‘60° angle’ and vice versa.
Are these two angles complementary?
No
(i) (ii) (iii) (iv)
Are these two angles complementary?
Yes
Fig 5.4
2024-25
Page 3


MATHEMATICS 74
Lines and
Angles
5.1 INTRODUCTION
You already know how to identify different lines, line segments and angles in a given
shape. Can you identify the different line segments and angles formed in the following
figures? (Fig 5.1)
(i) (ii) (iii) (iv)
Fig 5.1
Can you also identify whether the angles made are acute or obtuse or right?
Recall that a line segment has two end points. If we extend the two end points in either
direction endlessly, we get a line. Thus, we can say that a line has no end points. On the other
hand, recall that a ray has one end point (namely its starting point). For example, look at the
figures given below:
Fig 5.2
Here, Fig 5.2 (i) shows a line segment, Fig 5.2 (ii) shows a line and Fig 5.2 (iii) is that
of a ray. A line segment PQ is generally denoted by the symbol 
PQ
, a line AB is denoted by
the symbol AB

 
and the ray OP is denoted by OP
ur uu
. Give some examples of line segments and
rays from your daily life and discuss them with your friends.
(iii)
(i)
(ii)
Chapter  5
2024-25
LINES AND ANGLES 75
Again recall that an angle is formed when lines or line segments meet. In Fig 5.1,
observe the corners. These corners are formed when two lines or line segments intersect
at a point. For example, look at the figures given below:
Fig 5.3
In Fig 5.3 (i) line segments AB and BC intersect at B to form angle
ABC, and again line segments BC and AC intersect at C to form angle
ACB and so on. Whereas, in Fig 5.3 (ii) lines PQ and RS intersect at O
to form four angles POS, SOQ, QOR and ROP. An angle ABC is
represented by the symbol ?ABC. Thus, in Fig 5.3 (i), the three angles
formed are ?ABC, ?BCA and ?BAC, and in Fig 5.3 (ii), the four
angles formed are ? POS, ? SOQ, ? QOR and ? POR. Y ou have already
studied how to classify the angles as acute, obtuse or right angle.
Note: While referring to the measure of an angle ABC, we shall write m? ABC as simply
? ABC. The context will make it clear, whether we are referring to the angle or its
measure.
5.2 RELATED ANGLES
5.2.1  Complementary Angles
When the sum of the measures of two angles is 90°, the angles are called complementary
angles.
(i) (ii)
List ten figures around you
and identify the acute, obtuse
and right angles found in them.
TRY THESE
Whenever two angles are complementary , each angle is said to be the complement
of the other angle. In the above diagram (Fig 5.4), the ‘30° angle’ is the complement of the
‘60° angle’ and vice versa.
Are these two angles complementary?
No
(i) (ii) (iii) (iv)
Are these two angles complementary?
Yes
Fig 5.4
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MATHEMATICS 76
THINK, DISCUSS AND WRITE
1. Can two acute angles be complement to each other?
2. Can two obtuse angles be complement to each other?
3. Can two right angles be complement to each other?
TRY THESE
(i) (ii)
(iii) (iv)
  1. Which pairs of following angles are complementary? (Fig 5.5)
 Fig 5.5
  2. What is the measure of the complement of each of the following angles?
(i) 45º (ii) 65º (iii) 41º (iv) 54º
  3. The difference in the measures of two complementary angles is 12
o
. Find the measures of
the angles.
(i) (ii)
5.2.2  Supplementary Angles
Let us now look at the following pairs of angles (Fig 5.6):
2024-25
Page 4


MATHEMATICS 74
Lines and
Angles
5.1 INTRODUCTION
You already know how to identify different lines, line segments and angles in a given
shape. Can you identify the different line segments and angles formed in the following
figures? (Fig 5.1)
(i) (ii) (iii) (iv)
Fig 5.1
Can you also identify whether the angles made are acute or obtuse or right?
Recall that a line segment has two end points. If we extend the two end points in either
direction endlessly, we get a line. Thus, we can say that a line has no end points. On the other
hand, recall that a ray has one end point (namely its starting point). For example, look at the
figures given below:
Fig 5.2
Here, Fig 5.2 (i) shows a line segment, Fig 5.2 (ii) shows a line and Fig 5.2 (iii) is that
of a ray. A line segment PQ is generally denoted by the symbol 
PQ
, a line AB is denoted by
the symbol AB

 
and the ray OP is denoted by OP
ur uu
. Give some examples of line segments and
rays from your daily life and discuss them with your friends.
(iii)
(i)
(ii)
Chapter  5
2024-25
LINES AND ANGLES 75
Again recall that an angle is formed when lines or line segments meet. In Fig 5.1,
observe the corners. These corners are formed when two lines or line segments intersect
at a point. For example, look at the figures given below:
Fig 5.3
In Fig 5.3 (i) line segments AB and BC intersect at B to form angle
ABC, and again line segments BC and AC intersect at C to form angle
ACB and so on. Whereas, in Fig 5.3 (ii) lines PQ and RS intersect at O
to form four angles POS, SOQ, QOR and ROP. An angle ABC is
represented by the symbol ?ABC. Thus, in Fig 5.3 (i), the three angles
formed are ?ABC, ?BCA and ?BAC, and in Fig 5.3 (ii), the four
angles formed are ? POS, ? SOQ, ? QOR and ? POR. Y ou have already
studied how to classify the angles as acute, obtuse or right angle.
Note: While referring to the measure of an angle ABC, we shall write m? ABC as simply
? ABC. The context will make it clear, whether we are referring to the angle or its
measure.
5.2 RELATED ANGLES
5.2.1  Complementary Angles
When the sum of the measures of two angles is 90°, the angles are called complementary
angles.
(i) (ii)
List ten figures around you
and identify the acute, obtuse
and right angles found in them.
TRY THESE
Whenever two angles are complementary , each angle is said to be the complement
of the other angle. In the above diagram (Fig 5.4), the ‘30° angle’ is the complement of the
‘60° angle’ and vice versa.
Are these two angles complementary?
No
(i) (ii) (iii) (iv)
Are these two angles complementary?
Yes
Fig 5.4
2024-25
MATHEMATICS 76
THINK, DISCUSS AND WRITE
1. Can two acute angles be complement to each other?
2. Can two obtuse angles be complement to each other?
3. Can two right angles be complement to each other?
TRY THESE
(i) (ii)
(iii) (iv)
  1. Which pairs of following angles are complementary? (Fig 5.5)
 Fig 5.5
  2. What is the measure of the complement of each of the following angles?
(i) 45º (ii) 65º (iii) 41º (iv) 54º
  3. The difference in the measures of two complementary angles is 12
o
. Find the measures of
the angles.
(i) (ii)
5.2.2  Supplementary Angles
Let us now look at the following pairs of angles (Fig 5.6):
2024-25
LINES AND ANGLES 77
Fig 5.6
Do you notice that the sum of the measures of the angles in each of the above pairs
(Fig 5.6) comes out to be 180º? Such pairs of angles are called supplementary angles.
When two angles are supplementary, each angle is said to be the supplement of the other.
THINK, DISCUSS AND WRITE
1. Can two obtuse angles be supplementary?
2. Can two acute angles be supplementary?
3. Can two right angles be supplementary?
(iii) (iv)
TRY THESE
(iii) (iv)
(i) (ii)
1.Find the pairs of supplementary angles in Fig 5.7:
Fig 5.7
2024-25
Page 5


MATHEMATICS 74
Lines and
Angles
5.1 INTRODUCTION
You already know how to identify different lines, line segments and angles in a given
shape. Can you identify the different line segments and angles formed in the following
figures? (Fig 5.1)
(i) (ii) (iii) (iv)
Fig 5.1
Can you also identify whether the angles made are acute or obtuse or right?
Recall that a line segment has two end points. If we extend the two end points in either
direction endlessly, we get a line. Thus, we can say that a line has no end points. On the other
hand, recall that a ray has one end point (namely its starting point). For example, look at the
figures given below:
Fig 5.2
Here, Fig 5.2 (i) shows a line segment, Fig 5.2 (ii) shows a line and Fig 5.2 (iii) is that
of a ray. A line segment PQ is generally denoted by the symbol 
PQ
, a line AB is denoted by
the symbol AB

 
and the ray OP is denoted by OP
ur uu
. Give some examples of line segments and
rays from your daily life and discuss them with your friends.
(iii)
(i)
(ii)
Chapter  5
2024-25
LINES AND ANGLES 75
Again recall that an angle is formed when lines or line segments meet. In Fig 5.1,
observe the corners. These corners are formed when two lines or line segments intersect
at a point. For example, look at the figures given below:
Fig 5.3
In Fig 5.3 (i) line segments AB and BC intersect at B to form angle
ABC, and again line segments BC and AC intersect at C to form angle
ACB and so on. Whereas, in Fig 5.3 (ii) lines PQ and RS intersect at O
to form four angles POS, SOQ, QOR and ROP. An angle ABC is
represented by the symbol ?ABC. Thus, in Fig 5.3 (i), the three angles
formed are ?ABC, ?BCA and ?BAC, and in Fig 5.3 (ii), the four
angles formed are ? POS, ? SOQ, ? QOR and ? POR. Y ou have already
studied how to classify the angles as acute, obtuse or right angle.
Note: While referring to the measure of an angle ABC, we shall write m? ABC as simply
? ABC. The context will make it clear, whether we are referring to the angle or its
measure.
5.2 RELATED ANGLES
5.2.1  Complementary Angles
When the sum of the measures of two angles is 90°, the angles are called complementary
angles.
(i) (ii)
List ten figures around you
and identify the acute, obtuse
and right angles found in them.
TRY THESE
Whenever two angles are complementary , each angle is said to be the complement
of the other angle. In the above diagram (Fig 5.4), the ‘30° angle’ is the complement of the
‘60° angle’ and vice versa.
Are these two angles complementary?
No
(i) (ii) (iii) (iv)
Are these two angles complementary?
Yes
Fig 5.4
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MATHEMATICS 76
THINK, DISCUSS AND WRITE
1. Can two acute angles be complement to each other?
2. Can two obtuse angles be complement to each other?
3. Can two right angles be complement to each other?
TRY THESE
(i) (ii)
(iii) (iv)
  1. Which pairs of following angles are complementary? (Fig 5.5)
 Fig 5.5
  2. What is the measure of the complement of each of the following angles?
(i) 45º (ii) 65º (iii) 41º (iv) 54º
  3. The difference in the measures of two complementary angles is 12
o
. Find the measures of
the angles.
(i) (ii)
5.2.2  Supplementary Angles
Let us now look at the following pairs of angles (Fig 5.6):
2024-25
LINES AND ANGLES 77
Fig 5.6
Do you notice that the sum of the measures of the angles in each of the above pairs
(Fig 5.6) comes out to be 180º? Such pairs of angles are called supplementary angles.
When two angles are supplementary, each angle is said to be the supplement of the other.
THINK, DISCUSS AND WRITE
1. Can two obtuse angles be supplementary?
2. Can two acute angles be supplementary?
3. Can two right angles be supplementary?
(iii) (iv)
TRY THESE
(iii) (iv)
(i) (ii)
1.Find the pairs of supplementary angles in Fig 5.7:
Fig 5.7
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MATHEMATICS 78
EXERCISE 5.1
1. Find the complement of each of the following angles:
(i) (ii) (iii)
2. Find the supplement of each of the following angles:
(i) (ii) (iii)
3. Identify which of the following pairs of angles are complementary and which are
supplementary .
(i) 65º, 115º (ii) 63º,  27º (iii) 112º,  68º
(iv) 130º,  50º (v) 45º,  45º (vi) 80º,  10º
4. Find the angle which is equal to its complement.
5. Find the angle which is equal to its supplement.
6. In the given figure, ?1 and ?2 are supplementary
angles.
If ? 1 is decreased, what changes should take place
in ?2 so that both the angles still remain
supplementary.
7. Can two angles be supplementary if both of them are:
(i) acute? (ii) obtuse? (iii) right?
  2. What will be the measure of the supplement of each one of the following angles?
(i) 100º (ii) 90º (iii) 55º (iv) 125º
  3. Among two supplementary angles the measure of the larger angle is 44
o
 more
than the measure of the smaller. Find their measures.
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