NCERT Textbook: Parallel and Intersecting Lines

 Page 1


PARALLEL AND 
INTERSECTING 
LINES
5
5.1 Across the Line
Take a piece of square paper and fold it in di??erent ways. Now, on 
the creases formed by the folds, draw lines using a pencil and a scale. 
You will notice di??erent lines on the paper. Take any pair of lines and 
observe their relationship with each other. Do they meet? If they do 
not meet within the paper, do you think they would meet if they were 
extended beyond the paper?
Fig. 5.1
In this chapter, we will explore the relationship between lines on a 
plane surface. The table top, your piece of paper, the blackboard, and 
the bulletin board are all examples of plane surfaces.
Let us observe a pair of lines that meet each other. You will notice 
that they meet at a point. When a pair of lines meet each other at a 
point on a plane surface, we say that the lines intersect each other.
Let us observe what happens when two lines intersect.
How many angles do they form? 
In Fig. 5.2, where line l intersects line m, we can see that four angles 
are formed.
Chapter-5.indd   106 Chapter-5.indd   106 4/11/2025   7:30:37 PM 4/11/2025   7:30:37 PM
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Page 2


PARALLEL AND 
INTERSECTING 
LINES
5
5.1 Across the Line
Take a piece of square paper and fold it in di??erent ways. Now, on 
the creases formed by the folds, draw lines using a pencil and a scale. 
You will notice di??erent lines on the paper. Take any pair of lines and 
observe their relationship with each other. Do they meet? If they do 
not meet within the paper, do you think they would meet if they were 
extended beyond the paper?
Fig. 5.1
In this chapter, we will explore the relationship between lines on a 
plane surface. The table top, your piece of paper, the blackboard, and 
the bulletin board are all examples of plane surfaces.
Let us observe a pair of lines that meet each other. You will notice 
that they meet at a point. When a pair of lines meet each other at a 
point on a plane surface, we say that the lines intersect each other.
Let us observe what happens when two lines intersect.
How many angles do they form? 
In Fig. 5.2, where line l intersects line m, we can see that four angles 
are formed.
Chapter-5.indd   106 Chapter-5.indd   106 4/11/2025   7:30:37 PM 4/11/2025   7:30:37 PM
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Parallel and Intersecting Lines
m
a
b
l
c
d
Fig. 5.2
Can two straight lines intersect at more than one point?
Activity 1
Draw two lines on a plain sheet of paper so that they intersect. Measure 
the four angles formed with a protractor. Draw four such pairs of 
intersecting lines and measure the angles formed at the points of 
intersection.
What patterns do you observe among these angles?
In Fig. 5.2, if ?a is 120°, can you ??gure out the measurements of ?b, ?c 
and ?d, without drawing and measuring them?
We know that ?a and ?b together measure 180°, because when they 
are combined, they form a straight angle which measures 180°. So, if 
?a is 120°, then ?b must be 60°.
Similarly, ?b and ?c together measure 180°. So, if ?b is 60°, then ?c 
must be 120°. And ?c and ?d together measure 180°. So, if ?c is 120°, 
then ?d must be 60°.
Therefore, in Fig. 5.2, ?a and ?c measure 120°, and ?b and ?d 
measure 60°.
When two lines intersect each other and form four angles, labelled 
a, b, c and d, as in Fig. 5.2, then ?a and ?c are equal, and ?b and ?d 
are equal!
Is this always true for any pair of intersecting lines?
Check this for di??erent measures of ?a. Using these measurements, 
can you reason whether this property holds true for any measure of ?a?
We can generalise our reasoning for Fig. 5.2, without assuming the 
values of ?a.
 Since straight angles measure 180°, we must have ?a + ?b = ?a + ?d 
= 180°. Hence, ?b and ?d are always equal. Similarly, ?b + ?a = ?b + 
?c = 180°, so ?a and ?c must be equal.  
Adjacent angles, like ?a and ?b, formed by two lines intersecting 
each other, are called linear pairs. Linear pairs always add up to 180°.
107
Chapter-5.indd   107 Chapter-5.indd   107 04-Sep-25   3:39:15 PM 04-Sep-25   3:39:15 PM
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Page 3


PARALLEL AND 
INTERSECTING 
LINES
5
5.1 Across the Line
Take a piece of square paper and fold it in di??erent ways. Now, on 
the creases formed by the folds, draw lines using a pencil and a scale. 
You will notice di??erent lines on the paper. Take any pair of lines and 
observe their relationship with each other. Do they meet? If they do 
not meet within the paper, do you think they would meet if they were 
extended beyond the paper?
Fig. 5.1
In this chapter, we will explore the relationship between lines on a 
plane surface. The table top, your piece of paper, the blackboard, and 
the bulletin board are all examples of plane surfaces.
Let us observe a pair of lines that meet each other. You will notice 
that they meet at a point. When a pair of lines meet each other at a 
point on a plane surface, we say that the lines intersect each other.
Let us observe what happens when two lines intersect.
How many angles do they form? 
In Fig. 5.2, where line l intersects line m, we can see that four angles 
are formed.
Chapter-5.indd   106 Chapter-5.indd   106 4/11/2025   7:30:37 PM 4/11/2025   7:30:37 PM
Reprint 2026-27
Parallel and Intersecting Lines
m
a
b
l
c
d
Fig. 5.2
Can two straight lines intersect at more than one point?
Activity 1
Draw two lines on a plain sheet of paper so that they intersect. Measure 
the four angles formed with a protractor. Draw four such pairs of 
intersecting lines and measure the angles formed at the points of 
intersection.
What patterns do you observe among these angles?
In Fig. 5.2, if ?a is 120°, can you ??gure out the measurements of ?b, ?c 
and ?d, without drawing and measuring them?
We know that ?a and ?b together measure 180°, because when they 
are combined, they form a straight angle which measures 180°. So, if 
?a is 120°, then ?b must be 60°.
Similarly, ?b and ?c together measure 180°. So, if ?b is 60°, then ?c 
must be 120°. And ?c and ?d together measure 180°. So, if ?c is 120°, 
then ?d must be 60°.
Therefore, in Fig. 5.2, ?a and ?c measure 120°, and ?b and ?d 
measure 60°.
When two lines intersect each other and form four angles, labelled 
a, b, c and d, as in Fig. 5.2, then ?a and ?c are equal, and ?b and ?d 
are equal!
Is this always true for any pair of intersecting lines?
Check this for di??erent measures of ?a. Using these measurements, 
can you reason whether this property holds true for any measure of ?a?
We can generalise our reasoning for Fig. 5.2, without assuming the 
values of ?a.
 Since straight angles measure 180°, we must have ?a + ?b = ?a + ?d 
= 180°. Hence, ?b and ?d are always equal. Similarly, ?b + ?a = ?b + 
?c = 180°, so ?a and ?c must be equal.  
Adjacent angles, like ?a and ?b, formed by two lines intersecting 
each other, are called linear pairs. Linear pairs always add up to 180°.
107
Chapter-5.indd   107 Chapter-5.indd   107 04-Sep-25   3:39:15 PM 04-Sep-25   3:39:15 PM
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Ganita Prakash | Grade 7 
Opposite angles, like ?b and ?d, formed by two lines intersecting 
each other, are called vertically opposite angles. Vertically opposite 
angles are always equal to each other.
From the above reasoning, we conclude that whenever two lines 
intersect, vertically opposite angles are equal. Such a justi??cation is 
called a proof in mathematics.
Figure it Out
List all the linear pairs and vertically opposite angles you 
observe in Fig. 5.3:
Linear Pairs ?a and ?b, …
Pairs of Vertically 
Opposite Angles
?b and ?d, …
Measurements and Geometry
You might have noticed that when you measure linear pairs, sometimes 
they may not add up to 180°. Or, when you measure vertically opposite 
angles they may be unequal sometimes. What are the reasons for this?
There could be di??erent reasons:
• Measurement errors because of improper use of measuring 
instruments?—?in this case, a protractor
• Variation in the thickness of the lines drawn. The “ideal” line in 
geometry does not have any thickness! But it is not possible for us 
to draw lines without any thickness
In geometry, we create ideal versions of “lines” and other shapes 
we see around us, and analyse the relationships between them. For 
example, we know that the angle formed by a straight line is 180°. 
So, if another line divides this angle into two parts, both parts should 
add up to 180°. We arrive at this simply through reasoning and not 
by measurement. When we measure, it might not be exactly so, for 
the reasons mentioned above. Still the measurements come out very 
close to what we predict, because of which geometry ??nds widespread 
application in di??erent disciplines such as physics, art, engineering 
and architecture.
5.2 Perpendicular Lines
Can you draw a pair of intersecting lines such that all four angles are 
equal? Can you ??gure out what will be the measure of each angle ?
c
b
a
d
l
m
Fig. 5.3
108
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Page 4


PARALLEL AND 
INTERSECTING 
LINES
5
5.1 Across the Line
Take a piece of square paper and fold it in di??erent ways. Now, on 
the creases formed by the folds, draw lines using a pencil and a scale. 
You will notice di??erent lines on the paper. Take any pair of lines and 
observe their relationship with each other. Do they meet? If they do 
not meet within the paper, do you think they would meet if they were 
extended beyond the paper?
Fig. 5.1
In this chapter, we will explore the relationship between lines on a 
plane surface. The table top, your piece of paper, the blackboard, and 
the bulletin board are all examples of plane surfaces.
Let us observe a pair of lines that meet each other. You will notice 
that they meet at a point. When a pair of lines meet each other at a 
point on a plane surface, we say that the lines intersect each other.
Let us observe what happens when two lines intersect.
How many angles do they form? 
In Fig. 5.2, where line l intersects line m, we can see that four angles 
are formed.
Chapter-5.indd   106 Chapter-5.indd   106 4/11/2025   7:30:37 PM 4/11/2025   7:30:37 PM
Reprint 2026-27
Parallel and Intersecting Lines
m
a
b
l
c
d
Fig. 5.2
Can two straight lines intersect at more than one point?
Activity 1
Draw two lines on a plain sheet of paper so that they intersect. Measure 
the four angles formed with a protractor. Draw four such pairs of 
intersecting lines and measure the angles formed at the points of 
intersection.
What patterns do you observe among these angles?
In Fig. 5.2, if ?a is 120°, can you ??gure out the measurements of ?b, ?c 
and ?d, without drawing and measuring them?
We know that ?a and ?b together measure 180°, because when they 
are combined, they form a straight angle which measures 180°. So, if 
?a is 120°, then ?b must be 60°.
Similarly, ?b and ?c together measure 180°. So, if ?b is 60°, then ?c 
must be 120°. And ?c and ?d together measure 180°. So, if ?c is 120°, 
then ?d must be 60°.
Therefore, in Fig. 5.2, ?a and ?c measure 120°, and ?b and ?d 
measure 60°.
When two lines intersect each other and form four angles, labelled 
a, b, c and d, as in Fig. 5.2, then ?a and ?c are equal, and ?b and ?d 
are equal!
Is this always true for any pair of intersecting lines?
Check this for di??erent measures of ?a. Using these measurements, 
can you reason whether this property holds true for any measure of ?a?
We can generalise our reasoning for Fig. 5.2, without assuming the 
values of ?a.
 Since straight angles measure 180°, we must have ?a + ?b = ?a + ?d 
= 180°. Hence, ?b and ?d are always equal. Similarly, ?b + ?a = ?b + 
?c = 180°, so ?a and ?c must be equal.  
Adjacent angles, like ?a and ?b, formed by two lines intersecting 
each other, are called linear pairs. Linear pairs always add up to 180°.
107
Chapter-5.indd   107 Chapter-5.indd   107 04-Sep-25   3:39:15 PM 04-Sep-25   3:39:15 PM
Reprint 2026-27
Ganita Prakash | Grade 7 
Opposite angles, like ?b and ?d, formed by two lines intersecting 
each other, are called vertically opposite angles. Vertically opposite 
angles are always equal to each other.
From the above reasoning, we conclude that whenever two lines 
intersect, vertically opposite angles are equal. Such a justi??cation is 
called a proof in mathematics.
Figure it Out
List all the linear pairs and vertically opposite angles you 
observe in Fig. 5.3:
Linear Pairs ?a and ?b, …
Pairs of Vertically 
Opposite Angles
?b and ?d, …
Measurements and Geometry
You might have noticed that when you measure linear pairs, sometimes 
they may not add up to 180°. Or, when you measure vertically opposite 
angles they may be unequal sometimes. What are the reasons for this?
There could be di??erent reasons:
• Measurement errors because of improper use of measuring 
instruments?—?in this case, a protractor
• Variation in the thickness of the lines drawn. The “ideal” line in 
geometry does not have any thickness! But it is not possible for us 
to draw lines without any thickness
In geometry, we create ideal versions of “lines” and other shapes 
we see around us, and analyse the relationships between them. For 
example, we know that the angle formed by a straight line is 180°. 
So, if another line divides this angle into two parts, both parts should 
add up to 180°. We arrive at this simply through reasoning and not 
by measurement. When we measure, it might not be exactly so, for 
the reasons mentioned above. Still the measurements come out very 
close to what we predict, because of which geometry ??nds widespread 
application in di??erent disciplines such as physics, art, engineering 
and architecture.
5.2 Perpendicular Lines
Can you draw a pair of intersecting lines such that all four angles are 
equal? Can you ??gure out what will be the measure of each angle ?
c
b
a
d
l
m
Fig. 5.3
108
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Parallel and Intersecting Lines
m
l
Fig. 5.4
If two lines intersect and all four angles are equal, then each angle 
must be a right angle (90°).
Perpendicular lines are a pair of lines which intersect each other 
at right angles (90°). In Fig. 5.4, we can say that lines l and m are 
perpendicular to each other.
5.3 Between Lines
Observe Fig. 5.5 and describe the way the line segments meet or cross 
each other in each case, with appropriate mathematical words (a point, 
an endpoint, the midpoint, meet, intersect) and the degree measure of 
each angle.
For example, line segments FG and FH meet at the endpoint F at an 
angle 115.3°.
S
T
V
U
C
A
B
D
X
H
M
J
Y
I
L
R
P
O
Q
F
G
115.3°
Fig. 5.5
Are line segments ST and UV likely to meet if they are extended?
Are line segments OP and QR likely to meet if they are extended?
Here are some examples of lines we notice around us.
109
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Page 5


PARALLEL AND 
INTERSECTING 
LINES
5
5.1 Across the Line
Take a piece of square paper and fold it in di??erent ways. Now, on 
the creases formed by the folds, draw lines using a pencil and a scale. 
You will notice di??erent lines on the paper. Take any pair of lines and 
observe their relationship with each other. Do they meet? If they do 
not meet within the paper, do you think they would meet if they were 
extended beyond the paper?
Fig. 5.1
In this chapter, we will explore the relationship between lines on a 
plane surface. The table top, your piece of paper, the blackboard, and 
the bulletin board are all examples of plane surfaces.
Let us observe a pair of lines that meet each other. You will notice 
that they meet at a point. When a pair of lines meet each other at a 
point on a plane surface, we say that the lines intersect each other.
Let us observe what happens when two lines intersect.
How many angles do they form? 
In Fig. 5.2, where line l intersects line m, we can see that four angles 
are formed.
Chapter-5.indd   106 Chapter-5.indd   106 4/11/2025   7:30:37 PM 4/11/2025   7:30:37 PM
Reprint 2026-27
Parallel and Intersecting Lines
m
a
b
l
c
d
Fig. 5.2
Can two straight lines intersect at more than one point?
Activity 1
Draw two lines on a plain sheet of paper so that they intersect. Measure 
the four angles formed with a protractor. Draw four such pairs of 
intersecting lines and measure the angles formed at the points of 
intersection.
What patterns do you observe among these angles?
In Fig. 5.2, if ?a is 120°, can you ??gure out the measurements of ?b, ?c 
and ?d, without drawing and measuring them?
We know that ?a and ?b together measure 180°, because when they 
are combined, they form a straight angle which measures 180°. So, if 
?a is 120°, then ?b must be 60°.
Similarly, ?b and ?c together measure 180°. So, if ?b is 60°, then ?c 
must be 120°. And ?c and ?d together measure 180°. So, if ?c is 120°, 
then ?d must be 60°.
Therefore, in Fig. 5.2, ?a and ?c measure 120°, and ?b and ?d 
measure 60°.
When two lines intersect each other and form four angles, labelled 
a, b, c and d, as in Fig. 5.2, then ?a and ?c are equal, and ?b and ?d 
are equal!
Is this always true for any pair of intersecting lines?
Check this for di??erent measures of ?a. Using these measurements, 
can you reason whether this property holds true for any measure of ?a?
We can generalise our reasoning for Fig. 5.2, without assuming the 
values of ?a.
 Since straight angles measure 180°, we must have ?a + ?b = ?a + ?d 
= 180°. Hence, ?b and ?d are always equal. Similarly, ?b + ?a = ?b + 
?c = 180°, so ?a and ?c must be equal.  
Adjacent angles, like ?a and ?b, formed by two lines intersecting 
each other, are called linear pairs. Linear pairs always add up to 180°.
107
Chapter-5.indd   107 Chapter-5.indd   107 04-Sep-25   3:39:15 PM 04-Sep-25   3:39:15 PM
Reprint 2026-27
Ganita Prakash | Grade 7 
Opposite angles, like ?b and ?d, formed by two lines intersecting 
each other, are called vertically opposite angles. Vertically opposite 
angles are always equal to each other.
From the above reasoning, we conclude that whenever two lines 
intersect, vertically opposite angles are equal. Such a justi??cation is 
called a proof in mathematics.
Figure it Out
List all the linear pairs and vertically opposite angles you 
observe in Fig. 5.3:
Linear Pairs ?a and ?b, …
Pairs of Vertically 
Opposite Angles
?b and ?d, …
Measurements and Geometry
You might have noticed that when you measure linear pairs, sometimes 
they may not add up to 180°. Or, when you measure vertically opposite 
angles they may be unequal sometimes. What are the reasons for this?
There could be di??erent reasons:
• Measurement errors because of improper use of measuring 
instruments?—?in this case, a protractor
• Variation in the thickness of the lines drawn. The “ideal” line in 
geometry does not have any thickness! But it is not possible for us 
to draw lines without any thickness
In geometry, we create ideal versions of “lines” and other shapes 
we see around us, and analyse the relationships between them. For 
example, we know that the angle formed by a straight line is 180°. 
So, if another line divides this angle into two parts, both parts should 
add up to 180°. We arrive at this simply through reasoning and not 
by measurement. When we measure, it might not be exactly so, for 
the reasons mentioned above. Still the measurements come out very 
close to what we predict, because of which geometry ??nds widespread 
application in di??erent disciplines such as physics, art, engineering 
and architecture.
5.2 Perpendicular Lines
Can you draw a pair of intersecting lines such that all four angles are 
equal? Can you ??gure out what will be the measure of each angle ?
c
b
a
d
l
m
Fig. 5.3
108
Chapter-5.indd   108 Chapter-5.indd   108 4/11/2025   7:30:37 PM 4/11/2025   7:30:37 PM
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Parallel and Intersecting Lines
m
l
Fig. 5.4
If two lines intersect and all four angles are equal, then each angle 
must be a right angle (90°).
Perpendicular lines are a pair of lines which intersect each other 
at right angles (90°). In Fig. 5.4, we can say that lines l and m are 
perpendicular to each other.
5.3 Between Lines
Observe Fig. 5.5 and describe the way the line segments meet or cross 
each other in each case, with appropriate mathematical words (a point, 
an endpoint, the midpoint, meet, intersect) and the degree measure of 
each angle.
For example, line segments FG and FH meet at the endpoint F at an 
angle 115.3°.
S
T
V
U
C
A
B
D
X
H
M
J
Y
I
L
R
P
O
Q
F
G
115.3°
Fig. 5.5
Are line segments ST and UV likely to meet if they are extended?
Are line segments OP and QR likely to meet if they are extended?
Here are some examples of lines we notice around us.
109
Chapter-5.indd   109 Chapter-5.indd   109 4/11/2025   7:30:37 PM 4/11/2025   7:30:37 PM
Reprint 2026-27
Ganita Prakash | Grade 7 
What is common to the lines in the pictures above? They do not seem 
likely to intersect each other. Such lines are called parallel lines.
Parallel lines are a pair of lines that lie on the same plane, and do 
not meet however far we extend them at both ends.
Name some parallel lines you can spot in your classroom.
Parallel lines are often used in artwork and shading.
Which pairs of lines appear to be parallel in Fig. 5.6 below?
a
c
d e
g
f
h
i
b
Fig. 5.6
Note to the Teacher: It is important that the lines lie on the same plane. A line 
drawn on a table and a line drawn on the board may never meet but that does 
not make them parallel. 
 
110
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