NCERT Textbook: Quadrilaterals | Mathematics Class 8- New NCERT (Ganita Prakash) PDF Download

 Page 1


In this chapter, we will study some interesting types of four-sided 
figures and solve problems based on them. Such figures are commonly 
known as quadrilaterals. The word ‘quadrilateral’ is derived from Latin 
words — quadri meaning four, and latus referring to sides.
Observe the following figures.
Figs. (i), (ii), and (iii) are quadrilaterals, and the others are not. Why?
The angles of a quadrilateral are the angles between its sides, as 
marked in Figs. (i), (ii), and (iii). 
We will start with the most familiar quadrilaterals — rectangles and 
squares. 
(i) (ii) (iii)
(iv) (v)
QUADRILATERALS
4
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Page 2


In this chapter, we will study some interesting types of four-sided 
figures and solve problems based on them. Such figures are commonly 
known as quadrilaterals. The word ‘quadrilateral’ is derived from Latin 
words — quadri meaning four, and latus referring to sides.
Observe the following figures.
Figs. (i), (ii), and (iii) are quadrilaterals, and the others are not. Why?
The angles of a quadrilateral are the angles between its sides, as 
marked in Figs. (i), (ii), and (iii). 
We will start with the most familiar quadrilaterals — rectangles and 
squares. 
(i) (ii) (iii)
(iv) (v)
QUADRILATERALS
4
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Quadrilaterals
83
4.1 Rectangles and Squares
We know what rectangles are. Let us define them.
Rectangle: A rectangle is a quadrilateral in which —
(i) The angles are all right angles (90°), and
(ii) The opposite sides are of equal length.
The definition precisely states the conditions a quadrilateral has to 
satisfy to be called a rectangle.
Are there other ways to define a rectangle?
Let us consider the following problem related to the construction of 
rectangles.
A Carpenter’s Problem
A carpenter needs to put together 
two thin strips of wood, as shown 
in Fig. 1, so that when a thread is 
passed through their endpoints, it 
forms a rectangle.
She already has one 8 cm long 
strip. What should be the length of 
the other strip? Where should they 
both be joined?
Let us first model the structure 
that the carpenter has to make. The 
strips can be modelled as line segments. They are the diagonals of the 
quadrilateral formed by their endpoints. For the quadrilateral to be a 
rectangle, we need to answer the following questions —
1.  What is the length of the other diagonal?
2.  What is the point of intersection of the two diagonals?
3.  What should the angle be between the diagonals?
Let us answer these questions using 
geometric reasoning (deduction). If 
that is challenging, try to construct/
measure some rectangles.
To find the answers to these 
questions, let us suppose that we 
have placed the diagonals such that 
their endpoints form the vertices of 
a rectangle, as shown in Fig. 2.
A
B
D
C
O
Fig. 1
A D
B C
Fig. 2
AC = 8 cm
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Page 3


In this chapter, we will study some interesting types of four-sided 
figures and solve problems based on them. Such figures are commonly 
known as quadrilaterals. The word ‘quadrilateral’ is derived from Latin 
words — quadri meaning four, and latus referring to sides.
Observe the following figures.
Figs. (i), (ii), and (iii) are quadrilaterals, and the others are not. Why?
The angles of a quadrilateral are the angles between its sides, as 
marked in Figs. (i), (ii), and (iii). 
We will start with the most familiar quadrilaterals — rectangles and 
squares. 
(i) (ii) (iii)
(iv) (v)
QUADRILATERALS
4
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Quadrilaterals
83
4.1 Rectangles and Squares
We know what rectangles are. Let us define them.
Rectangle: A rectangle is a quadrilateral in which —
(i) The angles are all right angles (90°), and
(ii) The opposite sides are of equal length.
The definition precisely states the conditions a quadrilateral has to 
satisfy to be called a rectangle.
Are there other ways to define a rectangle?
Let us consider the following problem related to the construction of 
rectangles.
A Carpenter’s Problem
A carpenter needs to put together 
two thin strips of wood, as shown 
in Fig. 1, so that when a thread is 
passed through their endpoints, it 
forms a rectangle.
She already has one 8 cm long 
strip. What should be the length of 
the other strip? Where should they 
both be joined?
Let us first model the structure 
that the carpenter has to make. The 
strips can be modelled as line segments. They are the diagonals of the 
quadrilateral formed by their endpoints. For the quadrilateral to be a 
rectangle, we need to answer the following questions —
1.  What is the length of the other diagonal?
2.  What is the point of intersection of the two diagonals?
3.  What should the angle be between the diagonals?
Let us answer these questions using 
geometric reasoning (deduction). If 
that is challenging, try to construct/
measure some rectangles.
To find the answers to these 
questions, let us suppose that we 
have placed the diagonals such that 
their endpoints form the vertices of 
a rectangle, as shown in Fig. 2.
A
B
D
C
O
Fig. 1
A D
B C
Fig. 2
AC = 8 cm
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Ganita Prakash | Grade 8 
84
Deduction 1 — What is the length of the other diagonal?
This can be deduced using congruence as follows —
Since ABCD is a rectangle, we have 
AB = CD
?BAD = ?CDA = 90°
AD is common to both triangles.
So, ?ADC ? ?DAB by the SAS congruence 
condition.
Therefore, AC = BD, since they are corresponding parts of congruent 
triangles. This shows that the diagonals of a rectangle always have the 
same length.
So the other diagonal must also be 8 cm long. You can verify this 
property by constructing/measuring some rectangles.
Deduction 2 — What is the point of intersection of the two diagonals?
This can also be found using congruence. Since we need to know the 
relation between OA and OC, and OB and OD, which two triangles of the 
rectangle ABCD should we consider?
Common side
A
B
D
C
The blue angles are equal since 
they are vertically opposite 
angles.
In order to show congruence, 
consider ?1 and ?2. Are they 
equal?
O
1
2
A
B
D
C
A
B
D
C
O
Since ?B = 90°, 
?3 + ?1 = 90°.
In ?BCD, since
?3 + ?2 + 90 = 180,
we have ?3 + ?2 = 90°.
1
3
O
A
B
D
C
2
3
O
A
B
D
C
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Page 4


In this chapter, we will study some interesting types of four-sided 
figures and solve problems based on them. Such figures are commonly 
known as quadrilaterals. The word ‘quadrilateral’ is derived from Latin 
words — quadri meaning four, and latus referring to sides.
Observe the following figures.
Figs. (i), (ii), and (iii) are quadrilaterals, and the others are not. Why?
The angles of a quadrilateral are the angles between its sides, as 
marked in Figs. (i), (ii), and (iii). 
We will start with the most familiar quadrilaterals — rectangles and 
squares. 
(i) (ii) (iii)
(iv) (v)
QUADRILATERALS
4
Chapter 4 Quadrilaterals 06-07-2025.indd   82 Chapter 4 Quadrilaterals 06-07-2025.indd   82 7/10/2025   3:19:32 PM 7/10/2025   3:19:32 PM
Quadrilaterals
83
4.1 Rectangles and Squares
We know what rectangles are. Let us define them.
Rectangle: A rectangle is a quadrilateral in which —
(i) The angles are all right angles (90°), and
(ii) The opposite sides are of equal length.
The definition precisely states the conditions a quadrilateral has to 
satisfy to be called a rectangle.
Are there other ways to define a rectangle?
Let us consider the following problem related to the construction of 
rectangles.
A Carpenter’s Problem
A carpenter needs to put together 
two thin strips of wood, as shown 
in Fig. 1, so that when a thread is 
passed through their endpoints, it 
forms a rectangle.
She already has one 8 cm long 
strip. What should be the length of 
the other strip? Where should they 
both be joined?
Let us first model the structure 
that the carpenter has to make. The 
strips can be modelled as line segments. They are the diagonals of the 
quadrilateral formed by their endpoints. For the quadrilateral to be a 
rectangle, we need to answer the following questions —
1.  What is the length of the other diagonal?
2.  What is the point of intersection of the two diagonals?
3.  What should the angle be between the diagonals?
Let us answer these questions using 
geometric reasoning (deduction). If 
that is challenging, try to construct/
measure some rectangles.
To find the answers to these 
questions, let us suppose that we 
have placed the diagonals such that 
their endpoints form the vertices of 
a rectangle, as shown in Fig. 2.
A
B
D
C
O
Fig. 1
A D
B C
Fig. 2
AC = 8 cm
Chapter 4 Quadrilaterals 06-07-2025.indd   83 Chapter 4 Quadrilaterals 06-07-2025.indd   83 7/10/2025   3:19:32 PM 7/10/2025   3:19:32 PM
Ganita Prakash | Grade 8 
84
Deduction 1 — What is the length of the other diagonal?
This can be deduced using congruence as follows —
Since ABCD is a rectangle, we have 
AB = CD
?BAD = ?CDA = 90°
AD is common to both triangles.
So, ?ADC ? ?DAB by the SAS congruence 
condition.
Therefore, AC = BD, since they are corresponding parts of congruent 
triangles. This shows that the diagonals of a rectangle always have the 
same length.
So the other diagonal must also be 8 cm long. You can verify this 
property by constructing/measuring some rectangles.
Deduction 2 — What is the point of intersection of the two diagonals?
This can also be found using congruence. Since we need to know the 
relation between OA and OC, and OB and OD, which two triangles of the 
rectangle ABCD should we consider?
Common side
A
B
D
C
The blue angles are equal since 
they are vertically opposite 
angles.
In order to show congruence, 
consider ?1 and ?2. Are they 
equal?
O
1
2
A
B
D
C
A
B
D
C
O
Since ?B = 90°, 
?3 + ?1 = 90°.
In ?BCD, since
?3 + ?2 + 90 = 180,
we have ?3 + ?2 = 90°.
1
3
O
A
B
D
C
2
3
O
A
B
D
C
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Quadrilaterals
85
So, ?1 = ?2 (= 90° – ?3).
Thus, by the AAS condition for congruence, ?AOB ? ?COD.
Hence OA = OC and OB = OD, since they are corresponding parts of 
congruent triangles. So, O is the midpoint of AC and BD.
This shows that the diagonals of a rectangle always intersect at 
their midpoints.
Therefore, to get a rectangle, the diagonals must be drawn so that 
they are equal and intersect at their midpoints.
When the diagonals cross at their midpoints, we say that the diagonals bisect 
each other. Bisecting a quantity means dividing it into two equal parts.
Verify this property by constructing some rectangles and measuring 
their diagonals and the points of intersection.
Can the following equalities be used to establish that ?AOD ? ?COB?
AO = CO (proved above)
?AOB = ?COD (vertically opposite angles)
AD = CB 
Deduction 3 — What are the angles between the diagonals?
Let us check what quadrilateral we get if we draw the 
two diagonals such that their lengths are equal, they 
bisect each other and have an arbitrary angle, say 
60°, between them as shown in the figure to the right.
Can you find all the remaining angles?  
Math 
Talk
60°
A B
D C
O
We can find the remaining angles between the 
diagonals using our understanding of vertically 
opposite angles and linear pairs.
O
60°
60°
120° 120°
A B
D C
In ?AOB, since OA = OB, the angles opposite them 
are equal, say a.
Can you find the value of a? 
O
a a
A B
D C
60°
60°
120° 120°
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Page 5


In this chapter, we will study some interesting types of four-sided 
figures and solve problems based on them. Such figures are commonly 
known as quadrilaterals. The word ‘quadrilateral’ is derived from Latin 
words — quadri meaning four, and latus referring to sides.
Observe the following figures.
Figs. (i), (ii), and (iii) are quadrilaterals, and the others are not. Why?
The angles of a quadrilateral are the angles between its sides, as 
marked in Figs. (i), (ii), and (iii). 
We will start with the most familiar quadrilaterals — rectangles and 
squares. 
(i) (ii) (iii)
(iv) (v)
QUADRILATERALS
4
Chapter 4 Quadrilaterals 06-07-2025.indd   82 Chapter 4 Quadrilaterals 06-07-2025.indd   82 7/10/2025   3:19:32 PM 7/10/2025   3:19:32 PM
Quadrilaterals
83
4.1 Rectangles and Squares
We know what rectangles are. Let us define them.
Rectangle: A rectangle is a quadrilateral in which —
(i) The angles are all right angles (90°), and
(ii) The opposite sides are of equal length.
The definition precisely states the conditions a quadrilateral has to 
satisfy to be called a rectangle.
Are there other ways to define a rectangle?
Let us consider the following problem related to the construction of 
rectangles.
A Carpenter’s Problem
A carpenter needs to put together 
two thin strips of wood, as shown 
in Fig. 1, so that when a thread is 
passed through their endpoints, it 
forms a rectangle.
She already has one 8 cm long 
strip. What should be the length of 
the other strip? Where should they 
both be joined?
Let us first model the structure 
that the carpenter has to make. The 
strips can be modelled as line segments. They are the diagonals of the 
quadrilateral formed by their endpoints. For the quadrilateral to be a 
rectangle, we need to answer the following questions —
1.  What is the length of the other diagonal?
2.  What is the point of intersection of the two diagonals?
3.  What should the angle be between the diagonals?
Let us answer these questions using 
geometric reasoning (deduction). If 
that is challenging, try to construct/
measure some rectangles.
To find the answers to these 
questions, let us suppose that we 
have placed the diagonals such that 
their endpoints form the vertices of 
a rectangle, as shown in Fig. 2.
A
B
D
C
O
Fig. 1
A D
B C
Fig. 2
AC = 8 cm
Chapter 4 Quadrilaterals 06-07-2025.indd   83 Chapter 4 Quadrilaterals 06-07-2025.indd   83 7/10/2025   3:19:32 PM 7/10/2025   3:19:32 PM
Ganita Prakash | Grade 8 
84
Deduction 1 — What is the length of the other diagonal?
This can be deduced using congruence as follows —
Since ABCD is a rectangle, we have 
AB = CD
?BAD = ?CDA = 90°
AD is common to both triangles.
So, ?ADC ? ?DAB by the SAS congruence 
condition.
Therefore, AC = BD, since they are corresponding parts of congruent 
triangles. This shows that the diagonals of a rectangle always have the 
same length.
So the other diagonal must also be 8 cm long. You can verify this 
property by constructing/measuring some rectangles.
Deduction 2 — What is the point of intersection of the two diagonals?
This can also be found using congruence. Since we need to know the 
relation between OA and OC, and OB and OD, which two triangles of the 
rectangle ABCD should we consider?
Common side
A
B
D
C
The blue angles are equal since 
they are vertically opposite 
angles.
In order to show congruence, 
consider ?1 and ?2. Are they 
equal?
O
1
2
A
B
D
C
A
B
D
C
O
Since ?B = 90°, 
?3 + ?1 = 90°.
In ?BCD, since
?3 + ?2 + 90 = 180,
we have ?3 + ?2 = 90°.
1
3
O
A
B
D
C
2
3
O
A
B
D
C
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Quadrilaterals
85
So, ?1 = ?2 (= 90° – ?3).
Thus, by the AAS condition for congruence, ?AOB ? ?COD.
Hence OA = OC and OB = OD, since they are corresponding parts of 
congruent triangles. So, O is the midpoint of AC and BD.
This shows that the diagonals of a rectangle always intersect at 
their midpoints.
Therefore, to get a rectangle, the diagonals must be drawn so that 
they are equal and intersect at their midpoints.
When the diagonals cross at their midpoints, we say that the diagonals bisect 
each other. Bisecting a quantity means dividing it into two equal parts.
Verify this property by constructing some rectangles and measuring 
their diagonals and the points of intersection.
Can the following equalities be used to establish that ?AOD ? ?COB?
AO = CO (proved above)
?AOB = ?COD (vertically opposite angles)
AD = CB 
Deduction 3 — What are the angles between the diagonals?
Let us check what quadrilateral we get if we draw the 
two diagonals such that their lengths are equal, they 
bisect each other and have an arbitrary angle, say 
60°, between them as shown in the figure to the right.
Can you find all the remaining angles?  
Math 
Talk
60°
A B
D C
O
We can find the remaining angles between the 
diagonals using our understanding of vertically 
opposite angles and linear pairs.
O
60°
60°
120° 120°
A B
D C
In ?AOB, since OA = OB, the angles opposite them 
are equal, say a.
Can you find the value of a? 
O
a a
A B
D C
60°
60°
120° 120°
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Ganita Prakash | Grade 8 
86
In ?AOB, we have, 
a + a + 60 = 180 (interior angles of a triangle).
Therefore 2a = 120.
Thus a = 60.
Similarly, we can find the values of all the other angles.
Can we now identify what type of quadrilateral ABCD is? 
Notice that its angles all add up to 90° (30° + 60°). 
What can we say about its sides?
We can see that ?AOB ? ?COD and ?AOD ? ?COB. Hence, AB = CD, and 
AD = CB, since they are corresponding parts of congruent triangles. 
Therefore, ABCD is a rectangle since it satisfies the definition of a 
rectangle.
Will ABCD remain a rectangle if the angles between the 
diagonals are changed? Can we generalise this? 
Take one of the angles between the diagonals as x. 
O
60°
60°
120° 120°
60° 60°
60° 60°
30° 30°
30°
30°
A B
D C
60°
60°
O
A B
D C
120° 120°
O
A B
D C
x
B
D C
A
O
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