NCERT Textbook: Lines and Angles

 Page 1


Lines and a ng Les
2
 In this chapter, we will explore some of the most basic ideas of 
geometry including points, lines, rays, line segments and angles. 
These ideas form the building blocks of ‘plane geometry’, and will 
help us in understanding more advanced topics in geometry such as 
the construction and analysis of different shapes.
 2.1 Point
Mark a dot on the paper with a sharp tip of a pencil. The sharper the 
tip, the thinner will be the dot. This tiny dot will give you an idea of 
a point. A point determines a precise location, but it has no length, 
breadth or height. Some models for a point are given below. 
The tip of a 
compass
The sharpened 
end of a pencil 
The pointed 
end of a needle 
If you mark three points on a piece of paper, 
you may be required to distinguish these three 
points. For this purpose, each of the three points 
may be denoted by a single capital letter such as 
P
Z
T
Chapter 2_Lines and Angles.indd   13 13-08-2024   16:14:25
Reprint 2026-27
Page 2


Lines and a ng Les
2
 In this chapter, we will explore some of the most basic ideas of 
geometry including points, lines, rays, line segments and angles. 
These ideas form the building blocks of ‘plane geometry’, and will 
help us in understanding more advanced topics in geometry such as 
the construction and analysis of different shapes.
 2.1 Point
Mark a dot on the paper with a sharp tip of a pencil. The sharper the 
tip, the thinner will be the dot. This tiny dot will give you an idea of 
a point. A point determines a precise location, but it has no length, 
breadth or height. Some models for a point are given below. 
The tip of a 
compass
The sharpened 
end of a pencil 
The pointed 
end of a needle 
If you mark three points on a piece of paper, 
you may be required to distinguish these three 
points. For this purpose, each of the three points 
may be denoted by a single capital letter such as 
P
Z
T
Chapter 2_Lines and Angles.indd   13 13-08-2024   16:14:25
Reprint 2026-27
Ganita Prakash | Grade 6
14
Z, P and T. These points are read as ‘Point Z’, ‘Point P’ and ‘Point T’. Of 
course, the dots represent precise locations and must be imagined to be 
invisibly thin.
 2.2 Line Segment
Fold a piece of paper and unfold it. Do you 
see a crease? This gives the idea of a line 
segment. It has two end points, A and B.
Mark any two points A and B on a sheet of 
paper. Try to connect A to B by various 
routes  (Fig. 2.1).
What is the shortest route from A to B? 
This shortest path from Point A to Point B 
(including A and B) as shown here is called 
the line segment from A to B. It is denoted by 
either AB or BA. The points A and B are called 
the end points of the line segment AB.
 2.3 Line
Imagine that the line segment from A to B (i.e., 
AB) is extended beyond A in one direction and 
beyond B in the other direction without any 
end (see Fig. 2.2). This is a model for a line. Do 
you think you can draw a complete picture of 
a line? No. Why?
A line through two points A and B is written as AB. It extends 
forever in both directions. Sometimes a line is denoted by a letter like 
l or m.
Observe that any two points determine a unique line that passes 
through both of them.
A
B
B
A
Fig. 2.1
A
B
m
Fig. 2.2
Chapter 2_Lines and Angles.indd   14 13-08-2024   16:14:25
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Page 3


Lines and a ng Les
2
 In this chapter, we will explore some of the most basic ideas of 
geometry including points, lines, rays, line segments and angles. 
These ideas form the building blocks of ‘plane geometry’, and will 
help us in understanding more advanced topics in geometry such as 
the construction and analysis of different shapes.
 2.1 Point
Mark a dot on the paper with a sharp tip of a pencil. The sharper the 
tip, the thinner will be the dot. This tiny dot will give you an idea of 
a point. A point determines a precise location, but it has no length, 
breadth or height. Some models for a point are given below. 
The tip of a 
compass
The sharpened 
end of a pencil 
The pointed 
end of a needle 
If you mark three points on a piece of paper, 
you may be required to distinguish these three 
points. For this purpose, each of the three points 
may be denoted by a single capital letter such as 
P
Z
T
Chapter 2_Lines and Angles.indd   13 13-08-2024   16:14:25
Reprint 2026-27
Ganita Prakash | Grade 6
14
Z, P and T. These points are read as ‘Point Z’, ‘Point P’ and ‘Point T’. Of 
course, the dots represent precise locations and must be imagined to be 
invisibly thin.
 2.2 Line Segment
Fold a piece of paper and unfold it. Do you 
see a crease? This gives the idea of a line 
segment. It has two end points, A and B.
Mark any two points A and B on a sheet of 
paper. Try to connect A to B by various 
routes  (Fig. 2.1).
What is the shortest route from A to B? 
This shortest path from Point A to Point B 
(including A and B) as shown here is called 
the line segment from A to B. It is denoted by 
either AB or BA. The points A and B are called 
the end points of the line segment AB.
 2.3 Line
Imagine that the line segment from A to B (i.e., 
AB) is extended beyond A in one direction and 
beyond B in the other direction without any 
end (see Fig. 2.2). This is a model for a line. Do 
you think you can draw a complete picture of 
a line? No. Why?
A line through two points A and B is written as AB. It extends 
forever in both directions. Sometimes a line is denoted by a letter like 
l or m.
Observe that any two points determine a unique line that passes 
through both of them.
A
B
B
A
Fig. 2.1
A
B
m
Fig. 2.2
Chapter 2_Lines and Angles.indd   14 13-08-2024   16:14:25
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Lines and Angles
15
2.4 Ray
A ray is a portion of a line that starts at one point (called the starting 
point or initial point of the ray) and goes on endlessly in a direction. 
The following are some models for a ray:
Beam of light from a 
lighthouse   
Ray of light from a torch Sun rays
Look at the diagram (Fig. 2.3) of a ray. Two points are 
marked on it. One is the starting point A and the other 
is a point P on the path of the ray. We then denote the 
ray by AP.
 Figure it Out
1. 
 
Can you help Rihan and Sheetal ??nd their answers?
A
P
Fig. 2.3
Rihan marked a point 
on a piece of paper. 
How many lines can he 
draw that pass through 
the point?
Sheetal marked two points 
on a piece of paper. How 
many different lines can 
she draw that pass through 
both of the points?
Chapter 2_Lines and Angles.indd   15 13-08-2024   16:14:25
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Page 4


Lines and a ng Les
2
 In this chapter, we will explore some of the most basic ideas of 
geometry including points, lines, rays, line segments and angles. 
These ideas form the building blocks of ‘plane geometry’, and will 
help us in understanding more advanced topics in geometry such as 
the construction and analysis of different shapes.
 2.1 Point
Mark a dot on the paper with a sharp tip of a pencil. The sharper the 
tip, the thinner will be the dot. This tiny dot will give you an idea of 
a point. A point determines a precise location, but it has no length, 
breadth or height. Some models for a point are given below. 
The tip of a 
compass
The sharpened 
end of a pencil 
The pointed 
end of a needle 
If you mark three points on a piece of paper, 
you may be required to distinguish these three 
points. For this purpose, each of the three points 
may be denoted by a single capital letter such as 
P
Z
T
Chapter 2_Lines and Angles.indd   13 13-08-2024   16:14:25
Reprint 2026-27
Ganita Prakash | Grade 6
14
Z, P and T. These points are read as ‘Point Z’, ‘Point P’ and ‘Point T’. Of 
course, the dots represent precise locations and must be imagined to be 
invisibly thin.
 2.2 Line Segment
Fold a piece of paper and unfold it. Do you 
see a crease? This gives the idea of a line 
segment. It has two end points, A and B.
Mark any two points A and B on a sheet of 
paper. Try to connect A to B by various 
routes  (Fig. 2.1).
What is the shortest route from A to B? 
This shortest path from Point A to Point B 
(including A and B) as shown here is called 
the line segment from A to B. It is denoted by 
either AB or BA. The points A and B are called 
the end points of the line segment AB.
 2.3 Line
Imagine that the line segment from A to B (i.e., 
AB) is extended beyond A in one direction and 
beyond B in the other direction without any 
end (see Fig. 2.2). This is a model for a line. Do 
you think you can draw a complete picture of 
a line? No. Why?
A line through two points A and B is written as AB. It extends 
forever in both directions. Sometimes a line is denoted by a letter like 
l or m.
Observe that any two points determine a unique line that passes 
through both of them.
A
B
B
A
Fig. 2.1
A
B
m
Fig. 2.2
Chapter 2_Lines and Angles.indd   14 13-08-2024   16:14:25
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Lines and Angles
15
2.4 Ray
A ray is a portion of a line that starts at one point (called the starting 
point or initial point of the ray) and goes on endlessly in a direction. 
The following are some models for a ray:
Beam of light from a 
lighthouse   
Ray of light from a torch Sun rays
Look at the diagram (Fig. 2.3) of a ray. Two points are 
marked on it. One is the starting point A and the other 
is a point P on the path of the ray. We then denote the 
ray by AP.
 Figure it Out
1. 
 
Can you help Rihan and Sheetal ??nd their answers?
A
P
Fig. 2.3
Rihan marked a point 
on a piece of paper. 
How many lines can he 
draw that pass through 
the point?
Sheetal marked two points 
on a piece of paper. How 
many different lines can 
she draw that pass through 
both of the points?
Chapter 2_Lines and Angles.indd   15 13-08-2024   16:14:25
Reprint 2026-27
Ganita Prakash | Grade 6
16
2. Name the line segments in Fig. 2.4. Which of the ??ve marked 
points are on exactly one of the line segments?  Which are on two 
of the line segments? 
L
M
P
Q
R
Fig. 2.4
3.  Name the rays shown in Fig. 2.5. Is T the starting point of each of 
these rays?
A
T
Fig. 2.5
N B
4. Draw a rough ??gure and write labels appropriately to illustrate 
each of the following:
a. OP and OQ meet at O.
b. XY  and PQ intersect at point M.
c.  Line l contains points E and F but not point D.
d. Point P lies on AB.  
5. In Fig. 2.6, name: 
a. Five points
b. A line
c. Four rays
d. Five line segments
D
E
O
C
B
Fig. 2.6
Chapter 2_Lines and Angles.indd   16 13-08-2024   16:14:25
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Page 5


Lines and a ng Les
2
 In this chapter, we will explore some of the most basic ideas of 
geometry including points, lines, rays, line segments and angles. 
These ideas form the building blocks of ‘plane geometry’, and will 
help us in understanding more advanced topics in geometry such as 
the construction and analysis of different shapes.
 2.1 Point
Mark a dot on the paper with a sharp tip of a pencil. The sharper the 
tip, the thinner will be the dot. This tiny dot will give you an idea of 
a point. A point determines a precise location, but it has no length, 
breadth or height. Some models for a point are given below. 
The tip of a 
compass
The sharpened 
end of a pencil 
The pointed 
end of a needle 
If you mark three points on a piece of paper, 
you may be required to distinguish these three 
points. For this purpose, each of the three points 
may be denoted by a single capital letter such as 
P
Z
T
Chapter 2_Lines and Angles.indd   13 13-08-2024   16:14:25
Reprint 2026-27
Ganita Prakash | Grade 6
14
Z, P and T. These points are read as ‘Point Z’, ‘Point P’ and ‘Point T’. Of 
course, the dots represent precise locations and must be imagined to be 
invisibly thin.
 2.2 Line Segment
Fold a piece of paper and unfold it. Do you 
see a crease? This gives the idea of a line 
segment. It has two end points, A and B.
Mark any two points A and B on a sheet of 
paper. Try to connect A to B by various 
routes  (Fig. 2.1).
What is the shortest route from A to B? 
This shortest path from Point A to Point B 
(including A and B) as shown here is called 
the line segment from A to B. It is denoted by 
either AB or BA. The points A and B are called 
the end points of the line segment AB.
 2.3 Line
Imagine that the line segment from A to B (i.e., 
AB) is extended beyond A in one direction and 
beyond B in the other direction without any 
end (see Fig. 2.2). This is a model for a line. Do 
you think you can draw a complete picture of 
a line? No. Why?
A line through two points A and B is written as AB. It extends 
forever in both directions. Sometimes a line is denoted by a letter like 
l or m.
Observe that any two points determine a unique line that passes 
through both of them.
A
B
B
A
Fig. 2.1
A
B
m
Fig. 2.2
Chapter 2_Lines and Angles.indd   14 13-08-2024   16:14:25
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Lines and Angles
15
2.4 Ray
A ray is a portion of a line that starts at one point (called the starting 
point or initial point of the ray) and goes on endlessly in a direction. 
The following are some models for a ray:
Beam of light from a 
lighthouse   
Ray of light from a torch Sun rays
Look at the diagram (Fig. 2.3) of a ray. Two points are 
marked on it. One is the starting point A and the other 
is a point P on the path of the ray. We then denote the 
ray by AP.
 Figure it Out
1. 
 
Can you help Rihan and Sheetal ??nd their answers?
A
P
Fig. 2.3
Rihan marked a point 
on a piece of paper. 
How many lines can he 
draw that pass through 
the point?
Sheetal marked two points 
on a piece of paper. How 
many different lines can 
she draw that pass through 
both of the points?
Chapter 2_Lines and Angles.indd   15 13-08-2024   16:14:25
Reprint 2026-27
Ganita Prakash | Grade 6
16
2. Name the line segments in Fig. 2.4. Which of the ??ve marked 
points are on exactly one of the line segments?  Which are on two 
of the line segments? 
L
M
P
Q
R
Fig. 2.4
3.  Name the rays shown in Fig. 2.5. Is T the starting point of each of 
these rays?
A
T
Fig. 2.5
N B
4. Draw a rough ??gure and write labels appropriately to illustrate 
each of the following:
a. OP and OQ meet at O.
b. XY  and PQ intersect at point M.
c.  Line l contains points E and F but not point D.
d. Point P lies on AB.  
5. In Fig. 2.6, name: 
a. Five points
b. A line
c. Four rays
d. Five line segments
D
E
O
C
B
Fig. 2.6
Chapter 2_Lines and Angles.indd   16 13-08-2024   16:14:25
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Lines and Angles
17
6. Here is a ray OA (Fig. 2.7). It starts at O and 
passes through the point A. It also passes 
through the point B.
a. Can you also name it as OB ? Why? 
b. Can we write OA as AO ? Why or why not? 
2.5 Angle
An angle is formed by two rays having a 
common starting point. Here is an angle 
formed by rays BD and BE where B is 
the common starting point (Fig. 2.8).
The point B is called the vertex of the 
angle, and the rays BD and BE are called 
the arms of the angle. How can we name 
this angle? We can simply use the vertex and say that it is the Angle B. 
To be clearer, we use a point on each of the arms together with the 
vertex to name the angle. In this case, we name the angle as Angle DBE 
or Angle EBD. The word angle can be replaced by the symbol ‘?’, i.e., 
?DBE or ?EBD. Note that in specifying the angle, the vertex is always 
written as the middle letter.
To indicate an angle, we use a small curve at the vertex (refer 
to Fig. 2.9).
Vidya has just opened her book. Let us observe her opening the 
cover of the book in different scenarios.
Case 1 Case 2 Case 3 Case 4 Case 5 Case 6
O
B
A
Fig. 2.7
Fig. 2.8
B
D
E
vertex
arm
arm
Chapter 2_Lines and Angles.indd   17 13-08-2024   16:14:27
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