NCERT Textbook: Lines & Angles | Mathematics (Maths) Class 9 PDF Download

 Page 1


CHAPTER 6
LINES AND ANGLES
6.1 Introduction
In Chapter 5, you have studied that a minimum of two points are required to draw a
line. You have also studied some axioms and, with the help of these axioms, you
proved some other statements. In this chapter, you will study the properties of the
angles formed when two lines intersect each other, and also the properties of the
angles formed when a line intersects two or more parallel lines at distinct points.
Further you will use these properties to prove some statements using deductive reasoning
(see Appendix 1). You have already verified these statements through some activities
in the earlier classes.
In your daily life, you see different types of angles formed between the edges of
plane surfaces. For making a similar kind of model using the plane surfaces, you need
to have a thorough knowledge of angles. For instance, suppose you want to make a
model of a hut to keep in the school exhibition using bamboo sticks. Imagine how you
would make it? You would keep some of the sticks parallel to each other, and some
sticks would be kept slanted. Whenever an architect has to draw a plan for a multistoried
building, she has to draw intersecting lines and parallel lines at different angles. Without
the knowledge of the properties of these lines and angles, do you think she can draw
the layout of the building?
In science, you study the properties of light by drawing the ray diagrams.
For example, to study the refraction property of light when it enters from one medium
to the other medium, you use the properties of intersecting lines and parallel lines.
When two or more forces act on a body, you draw the diagram in which forces are
represented by directed line segments to study the net effect of the forces on the
body. At that time, you need to know the relation between the angles when the rays
(or line segments) are parallel to or intersect each other. To find the height of a tower
or to find the distance of a ship from the light house, one needs to know the angle
2024-25
Page 2


CHAPTER 6
LINES AND ANGLES
6.1 Introduction
In Chapter 5, you have studied that a minimum of two points are required to draw a
line. You have also studied some axioms and, with the help of these axioms, you
proved some other statements. In this chapter, you will study the properties of the
angles formed when two lines intersect each other, and also the properties of the
angles formed when a line intersects two or more parallel lines at distinct points.
Further you will use these properties to prove some statements using deductive reasoning
(see Appendix 1). You have already verified these statements through some activities
in the earlier classes.
In your daily life, you see different types of angles formed between the edges of
plane surfaces. For making a similar kind of model using the plane surfaces, you need
to have a thorough knowledge of angles. For instance, suppose you want to make a
model of a hut to keep in the school exhibition using bamboo sticks. Imagine how you
would make it? You would keep some of the sticks parallel to each other, and some
sticks would be kept slanted. Whenever an architect has to draw a plan for a multistoried
building, she has to draw intersecting lines and parallel lines at different angles. Without
the knowledge of the properties of these lines and angles, do you think she can draw
the layout of the building?
In science, you study the properties of light by drawing the ray diagrams.
For example, to study the refraction property of light when it enters from one medium
to the other medium, you use the properties of intersecting lines and parallel lines.
When two or more forces act on a body, you draw the diagram in which forces are
represented by directed line segments to study the net effect of the forces on the
body. At that time, you need to know the relation between the angles when the rays
(or line segments) are parallel to or intersect each other. To find the height of a tower
or to find the distance of a ship from the light house, one needs to know the angle
2024-25
70 MATHEMA TICS
formed between the horizontal and the line of sight. Plenty of other examples can be
given where lines and angles are used. In the subsequent chapters of geometry, you
will be using these properties of lines and angles to deduce more and more useful
properties.
Let us first revise the terms and definitions related to lines and angles learnt in
earlier classes.
6.2 Basic Terms and Definitions
Recall that a part (or portion) of a line with two end points is called a line-segment
and a part of a line with one end point is called a ray. Note that the line segment AB is
denoted by AB , and its length is denoted by AB. The ray AB is denoted by 
AB



 
, and
a line is denoted by 
AB


 
. However, we will not use these symbols, and will denote
the line segment AB, ray AB, length AB and line AB by the same symbol, AB. The
meaning will be clear from the context. Sometimes small letters l, m, n, etc. will be
used to denote lines.
If three or more points lie on the same line, they are called collinear points;
otherwise they are called non-collinear points.
Recall that an angle is formed when two rays originate from the same end point.
The rays making an angle are called the arms of the angle and the end point is called
the vertex of the angle. You have studied different types of angles, such as acute
angle, right angle, obtuse angle, straight angle and reflex angle in earlier classes
(see Fig. 6.1).
(i) acute angle : 0° < x < 90° (ii) right angle : y = 90° (iii) obtuse angle : 90° < z < 180°
(iv) straight angle : s = 180° (v) reflex angle : 180° < t < 360°
Fig. 6.1 : Types of Angles
2024-25
Page 3


CHAPTER 6
LINES AND ANGLES
6.1 Introduction
In Chapter 5, you have studied that a minimum of two points are required to draw a
line. You have also studied some axioms and, with the help of these axioms, you
proved some other statements. In this chapter, you will study the properties of the
angles formed when two lines intersect each other, and also the properties of the
angles formed when a line intersects two or more parallel lines at distinct points.
Further you will use these properties to prove some statements using deductive reasoning
(see Appendix 1). You have already verified these statements through some activities
in the earlier classes.
In your daily life, you see different types of angles formed between the edges of
plane surfaces. For making a similar kind of model using the plane surfaces, you need
to have a thorough knowledge of angles. For instance, suppose you want to make a
model of a hut to keep in the school exhibition using bamboo sticks. Imagine how you
would make it? You would keep some of the sticks parallel to each other, and some
sticks would be kept slanted. Whenever an architect has to draw a plan for a multistoried
building, she has to draw intersecting lines and parallel lines at different angles. Without
the knowledge of the properties of these lines and angles, do you think she can draw
the layout of the building?
In science, you study the properties of light by drawing the ray diagrams.
For example, to study the refraction property of light when it enters from one medium
to the other medium, you use the properties of intersecting lines and parallel lines.
When two or more forces act on a body, you draw the diagram in which forces are
represented by directed line segments to study the net effect of the forces on the
body. At that time, you need to know the relation between the angles when the rays
(or line segments) are parallel to or intersect each other. To find the height of a tower
or to find the distance of a ship from the light house, one needs to know the angle
2024-25
70 MATHEMA TICS
formed between the horizontal and the line of sight. Plenty of other examples can be
given where lines and angles are used. In the subsequent chapters of geometry, you
will be using these properties of lines and angles to deduce more and more useful
properties.
Let us first revise the terms and definitions related to lines and angles learnt in
earlier classes.
6.2 Basic Terms and Definitions
Recall that a part (or portion) of a line with two end points is called a line-segment
and a part of a line with one end point is called a ray. Note that the line segment AB is
denoted by AB , and its length is denoted by AB. The ray AB is denoted by 
AB



 
, and
a line is denoted by 
AB


 
. However, we will not use these symbols, and will denote
the line segment AB, ray AB, length AB and line AB by the same symbol, AB. The
meaning will be clear from the context. Sometimes small letters l, m, n, etc. will be
used to denote lines.
If three or more points lie on the same line, they are called collinear points;
otherwise they are called non-collinear points.
Recall that an angle is formed when two rays originate from the same end point.
The rays making an angle are called the arms of the angle and the end point is called
the vertex of the angle. You have studied different types of angles, such as acute
angle, right angle, obtuse angle, straight angle and reflex angle in earlier classes
(see Fig. 6.1).
(i) acute angle : 0° < x < 90° (ii) right angle : y = 90° (iii) obtuse angle : 90° < z < 180°
(iv) straight angle : s = 180° (v) reflex angle : 180° < t < 360°
Fig. 6.1 : Types of Angles
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LINES AND ANGLES 71
An acute angle measures between 0° and 90°, whereas a right angle is exactly
equal to 90°. An angle greater than 90° but less than 180° is called an obtuse angle.
Also, recall that a straight angle is equal to 180°. An angle which is greater than 180°
but less than 360° is called a reflex angle. Further, two angles whose sum is 90° are
called complementary angles, and two angles whose sum is 180° are called
supplementary angles.
Y ou have also studied about adjacent angles
in the earlier classes (see Fig. 6.2). Two angles
are adjacent, if they have a common vertex, a
common arm and their non-common arms are
on different sides of the common arm. In
Fig. 6.2, ? ABD and ? DBC are adjacent
angles. Ray BD is their common arm and point
B is their common vertex. Ray BA and ray BC
are non common arms. Moreover, when two
angles are adjacent, then their sum is always
equal to the angle formed by the two non-
common arms. So, we can write
? ABC = ? ABD + ? DBC.
Note that ? ABC and ? ABD are not
adjacent angles. Why? Because their non-
common arms BD and BC lie on the same side
of the common arm BA.
If the non-common arms BA and BC in
Fig. 6.2, form a line then it will look like Fig. 6.3.
In this case, ? ABD and ? DBC are called
linear pair of angles.
You may also recall the vertically opposite
angles formed when two lines, say AB and CD,
intersect each other, say at the point O
(see Fig. 6.4). There are two pairs of vertically
opposite angles.
One pair is ?AOD and ?BOC. Can you
find the other pair?
Fig. 6.2 : Adjacent angles
Fig. 6.3 : Linear pair of angles
Fig. 6.4 : V ertically opposite
angles
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Page 4


CHAPTER 6
LINES AND ANGLES
6.1 Introduction
In Chapter 5, you have studied that a minimum of two points are required to draw a
line. You have also studied some axioms and, with the help of these axioms, you
proved some other statements. In this chapter, you will study the properties of the
angles formed when two lines intersect each other, and also the properties of the
angles formed when a line intersects two or more parallel lines at distinct points.
Further you will use these properties to prove some statements using deductive reasoning
(see Appendix 1). You have already verified these statements through some activities
in the earlier classes.
In your daily life, you see different types of angles formed between the edges of
plane surfaces. For making a similar kind of model using the plane surfaces, you need
to have a thorough knowledge of angles. For instance, suppose you want to make a
model of a hut to keep in the school exhibition using bamboo sticks. Imagine how you
would make it? You would keep some of the sticks parallel to each other, and some
sticks would be kept slanted. Whenever an architect has to draw a plan for a multistoried
building, she has to draw intersecting lines and parallel lines at different angles. Without
the knowledge of the properties of these lines and angles, do you think she can draw
the layout of the building?
In science, you study the properties of light by drawing the ray diagrams.
For example, to study the refraction property of light when it enters from one medium
to the other medium, you use the properties of intersecting lines and parallel lines.
When two or more forces act on a body, you draw the diagram in which forces are
represented by directed line segments to study the net effect of the forces on the
body. At that time, you need to know the relation between the angles when the rays
(or line segments) are parallel to or intersect each other. To find the height of a tower
or to find the distance of a ship from the light house, one needs to know the angle
2024-25
70 MATHEMA TICS
formed between the horizontal and the line of sight. Plenty of other examples can be
given where lines and angles are used. In the subsequent chapters of geometry, you
will be using these properties of lines and angles to deduce more and more useful
properties.
Let us first revise the terms and definitions related to lines and angles learnt in
earlier classes.
6.2 Basic Terms and Definitions
Recall that a part (or portion) of a line with two end points is called a line-segment
and a part of a line with one end point is called a ray. Note that the line segment AB is
denoted by AB , and its length is denoted by AB. The ray AB is denoted by 
AB



 
, and
a line is denoted by 
AB


 
. However, we will not use these symbols, and will denote
the line segment AB, ray AB, length AB and line AB by the same symbol, AB. The
meaning will be clear from the context. Sometimes small letters l, m, n, etc. will be
used to denote lines.
If three or more points lie on the same line, they are called collinear points;
otherwise they are called non-collinear points.
Recall that an angle is formed when two rays originate from the same end point.
The rays making an angle are called the arms of the angle and the end point is called
the vertex of the angle. You have studied different types of angles, such as acute
angle, right angle, obtuse angle, straight angle and reflex angle in earlier classes
(see Fig. 6.1).
(i) acute angle : 0° < x < 90° (ii) right angle : y = 90° (iii) obtuse angle : 90° < z < 180°
(iv) straight angle : s = 180° (v) reflex angle : 180° < t < 360°
Fig. 6.1 : Types of Angles
2024-25
LINES AND ANGLES 71
An acute angle measures between 0° and 90°, whereas a right angle is exactly
equal to 90°. An angle greater than 90° but less than 180° is called an obtuse angle.
Also, recall that a straight angle is equal to 180°. An angle which is greater than 180°
but less than 360° is called a reflex angle. Further, two angles whose sum is 90° are
called complementary angles, and two angles whose sum is 180° are called
supplementary angles.
Y ou have also studied about adjacent angles
in the earlier classes (see Fig. 6.2). Two angles
are adjacent, if they have a common vertex, a
common arm and their non-common arms are
on different sides of the common arm. In
Fig. 6.2, ? ABD and ? DBC are adjacent
angles. Ray BD is their common arm and point
B is their common vertex. Ray BA and ray BC
are non common arms. Moreover, when two
angles are adjacent, then their sum is always
equal to the angle formed by the two non-
common arms. So, we can write
? ABC = ? ABD + ? DBC.
Note that ? ABC and ? ABD are not
adjacent angles. Why? Because their non-
common arms BD and BC lie on the same side
of the common arm BA.
If the non-common arms BA and BC in
Fig. 6.2, form a line then it will look like Fig. 6.3.
In this case, ? ABD and ? DBC are called
linear pair of angles.
You may also recall the vertically opposite
angles formed when two lines, say AB and CD,
intersect each other, say at the point O
(see Fig. 6.4). There are two pairs of vertically
opposite angles.
One pair is ?AOD and ?BOC. Can you
find the other pair?
Fig. 6.2 : Adjacent angles
Fig. 6.3 : Linear pair of angles
Fig. 6.4 : V ertically opposite
angles
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72 MATHEMA TICS
6.3 Intersecting Lines and Non-intersecting Lines
Draw two different lines PQ and RS on a paper. You will see that you can draw them
in two different ways as shown in Fig. 6.5 (i) and Fig. 6.5 (ii).
(i) Intersecting lines (ii) Non-intersecting (parallel) lines
Fig. 6.5 : Different ways of drawing two lines
Recall the notion of a line, that it extends indefinitely in both directions. Lines PQ
and RS in Fig. 6.5 (i) are intersecting lines and in Fig. 6.5 (ii) are parallel lines. Note
that the lengths of the common perpendiculars at different points on these parallel
lines is the same. This equal length is called the distance between two parallel lines.
6.4 Pairs of Angles
In Section 6.2, you have learnt the definitions of
some of the pairs of angles such as
complementary angles, supplementary angles,
adjacent angles, linear pair of angles, etc. Can
you think of some relations between these
angles? Now, let us find out the relation between
the angles formed when a ray stands on a line.
Draw a figure in which a ray stands on a line as
shown in Fig. 6.6. Name the line as AB and the
ray as OC. What are the angles formed at the
point O? They are ? AOC, ? BOC and ? AOB.
Can we write ? AOC + ? BOC = ? AOB? (1)
Yes! (Why? Refer to adjacent angles in Section 6.2)
What is the measure of ? AOB? It is 180°. (Why?) (2)
From (1) and (2), can you say that ? AOC + ? BOC = 180°? Yes! (Why?)
From the above discussion, we can state the following Axiom:
Fig. 6.6 : Linear pair of angles
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Page 5


CHAPTER 6
LINES AND ANGLES
6.1 Introduction
In Chapter 5, you have studied that a minimum of two points are required to draw a
line. You have also studied some axioms and, with the help of these axioms, you
proved some other statements. In this chapter, you will study the properties of the
angles formed when two lines intersect each other, and also the properties of the
angles formed when a line intersects two or more parallel lines at distinct points.
Further you will use these properties to prove some statements using deductive reasoning
(see Appendix 1). You have already verified these statements through some activities
in the earlier classes.
In your daily life, you see different types of angles formed between the edges of
plane surfaces. For making a similar kind of model using the plane surfaces, you need
to have a thorough knowledge of angles. For instance, suppose you want to make a
model of a hut to keep in the school exhibition using bamboo sticks. Imagine how you
would make it? You would keep some of the sticks parallel to each other, and some
sticks would be kept slanted. Whenever an architect has to draw a plan for a multistoried
building, she has to draw intersecting lines and parallel lines at different angles. Without
the knowledge of the properties of these lines and angles, do you think she can draw
the layout of the building?
In science, you study the properties of light by drawing the ray diagrams.
For example, to study the refraction property of light when it enters from one medium
to the other medium, you use the properties of intersecting lines and parallel lines.
When two or more forces act on a body, you draw the diagram in which forces are
represented by directed line segments to study the net effect of the forces on the
body. At that time, you need to know the relation between the angles when the rays
(or line segments) are parallel to or intersect each other. To find the height of a tower
or to find the distance of a ship from the light house, one needs to know the angle
2024-25
70 MATHEMA TICS
formed between the horizontal and the line of sight. Plenty of other examples can be
given where lines and angles are used. In the subsequent chapters of geometry, you
will be using these properties of lines and angles to deduce more and more useful
properties.
Let us first revise the terms and definitions related to lines and angles learnt in
earlier classes.
6.2 Basic Terms and Definitions
Recall that a part (or portion) of a line with two end points is called a line-segment
and a part of a line with one end point is called a ray. Note that the line segment AB is
denoted by AB , and its length is denoted by AB. The ray AB is denoted by 
AB



 
, and
a line is denoted by 
AB


 
. However, we will not use these symbols, and will denote
the line segment AB, ray AB, length AB and line AB by the same symbol, AB. The
meaning will be clear from the context. Sometimes small letters l, m, n, etc. will be
used to denote lines.
If three or more points lie on the same line, they are called collinear points;
otherwise they are called non-collinear points.
Recall that an angle is formed when two rays originate from the same end point.
The rays making an angle are called the arms of the angle and the end point is called
the vertex of the angle. You have studied different types of angles, such as acute
angle, right angle, obtuse angle, straight angle and reflex angle in earlier classes
(see Fig. 6.1).
(i) acute angle : 0° < x < 90° (ii) right angle : y = 90° (iii) obtuse angle : 90° < z < 180°
(iv) straight angle : s = 180° (v) reflex angle : 180° < t < 360°
Fig. 6.1 : Types of Angles
2024-25
LINES AND ANGLES 71
An acute angle measures between 0° and 90°, whereas a right angle is exactly
equal to 90°. An angle greater than 90° but less than 180° is called an obtuse angle.
Also, recall that a straight angle is equal to 180°. An angle which is greater than 180°
but less than 360° is called a reflex angle. Further, two angles whose sum is 90° are
called complementary angles, and two angles whose sum is 180° are called
supplementary angles.
Y ou have also studied about adjacent angles
in the earlier classes (see Fig. 6.2). Two angles
are adjacent, if they have a common vertex, a
common arm and their non-common arms are
on different sides of the common arm. In
Fig. 6.2, ? ABD and ? DBC are adjacent
angles. Ray BD is their common arm and point
B is their common vertex. Ray BA and ray BC
are non common arms. Moreover, when two
angles are adjacent, then their sum is always
equal to the angle formed by the two non-
common arms. So, we can write
? ABC = ? ABD + ? DBC.
Note that ? ABC and ? ABD are not
adjacent angles. Why? Because their non-
common arms BD and BC lie on the same side
of the common arm BA.
If the non-common arms BA and BC in
Fig. 6.2, form a line then it will look like Fig. 6.3.
In this case, ? ABD and ? DBC are called
linear pair of angles.
You may also recall the vertically opposite
angles formed when two lines, say AB and CD,
intersect each other, say at the point O
(see Fig. 6.4). There are two pairs of vertically
opposite angles.
One pair is ?AOD and ?BOC. Can you
find the other pair?
Fig. 6.2 : Adjacent angles
Fig. 6.3 : Linear pair of angles
Fig. 6.4 : V ertically opposite
angles
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72 MATHEMA TICS
6.3 Intersecting Lines and Non-intersecting Lines
Draw two different lines PQ and RS on a paper. You will see that you can draw them
in two different ways as shown in Fig. 6.5 (i) and Fig. 6.5 (ii).
(i) Intersecting lines (ii) Non-intersecting (parallel) lines
Fig. 6.5 : Different ways of drawing two lines
Recall the notion of a line, that it extends indefinitely in both directions. Lines PQ
and RS in Fig. 6.5 (i) are intersecting lines and in Fig. 6.5 (ii) are parallel lines. Note
that the lengths of the common perpendiculars at different points on these parallel
lines is the same. This equal length is called the distance between two parallel lines.
6.4 Pairs of Angles
In Section 6.2, you have learnt the definitions of
some of the pairs of angles such as
complementary angles, supplementary angles,
adjacent angles, linear pair of angles, etc. Can
you think of some relations between these
angles? Now, let us find out the relation between
the angles formed when a ray stands on a line.
Draw a figure in which a ray stands on a line as
shown in Fig. 6.6. Name the line as AB and the
ray as OC. What are the angles formed at the
point O? They are ? AOC, ? BOC and ? AOB.
Can we write ? AOC + ? BOC = ? AOB? (1)
Yes! (Why? Refer to adjacent angles in Section 6.2)
What is the measure of ? AOB? It is 180°. (Why?) (2)
From (1) and (2), can you say that ? AOC + ? BOC = 180°? Yes! (Why?)
From the above discussion, we can state the following Axiom:
Fig. 6.6 : Linear pair of angles
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LINES AND ANGLES 73
Axiom 6.1 : If a ray stands on a line, then the sum of two adjacent angles so
formed is 180°.
Recall that when the sum of two adjacent angles is 180°, then they are called a
linear pair of angles.
In Axiom 6.1, it is given that ‘a ray stands on a line’. From this ‘given’, we have
concluded that ‘the sum of two adjacent angles so formed is 180°’. Can we write
Axiom 6.1 the other way? That is, take the ‘conclusion’ of Axiom 6.1 as ‘given’ and
the ‘given’ as the ‘conclusion’. So it becomes:
(A) If the sum of two adjacent angles is 180°, then a ray stands on a line (that is,
the non-common arms form a line).
Now you see that the Axiom 6.1 and statement (A) are in a sense the reverse of
each others. We call each as converse of the other. We do not know whether the
statement (A) is true or not. Let us check. Draw adjacent angles of different measures
as shown in Fig. 6.7. Keep the ruler along one of the non-common arms in each case.
Does the other non-common arm also lie along the ruler?
Fig. 6.7 : Adjacent angles with different measures
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