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

 Page 1


UNDERSTANDING QUADRILATERALS  21
3.1  Introduction
Y ou know that the paper is a model for a plane surface. When you join a number of
points without lifting a pencil from the paper (and without retracing any portion of the
drawing other than single points), you get a plane curve.
3.1.1  Convex and concave polygons
A simple closed curve made up of only line segments is called a polygon.
          
Curves that are polygons Curves that are not polygons
Here are some convex polygons and some concave polygons. (Fig 3.1)
Convex polygons Concave polygons
Can you find how these types of polygons differ from one another? Polygons that are
convex have no portions of their diagonals in their exteriors or any line segment joining any
two different points, in the interior of the polygon, lies wholly in the interior of it . Is this
true with concave polygons? Study the figures given. Then try to describe in your own
words what we mean by a convex polygon and what we mean by a concave polygon. Give
two rough sketches of each kind.
In our work in this class, we will be dealing with convex polygons only .
3.1.2  Regular and irregular polygons
A regular polygon is both ‘equiangular’ and ‘equilateral’. For example, a square has sides
of equal length and angles of equal measure. Hence it is a regular polygon. A rectangle is
CHAPTER
3
Understanding
Quadrilaterals
Fig 3.1
Reprint 2024-25
Page 2


UNDERSTANDING QUADRILATERALS  21
3.1  Introduction
Y ou know that the paper is a model for a plane surface. When you join a number of
points without lifting a pencil from the paper (and without retracing any portion of the
drawing other than single points), you get a plane curve.
3.1.1  Convex and concave polygons
A simple closed curve made up of only line segments is called a polygon.
          
Curves that are polygons Curves that are not polygons
Here are some convex polygons and some concave polygons. (Fig 3.1)
Convex polygons Concave polygons
Can you find how these types of polygons differ from one another? Polygons that are
convex have no portions of their diagonals in their exteriors or any line segment joining any
two different points, in the interior of the polygon, lies wholly in the interior of it . Is this
true with concave polygons? Study the figures given. Then try to describe in your own
words what we mean by a convex polygon and what we mean by a concave polygon. Give
two rough sketches of each kind.
In our work in this class, we will be dealing with convex polygons only .
3.1.2  Regular and irregular polygons
A regular polygon is both ‘equiangular’ and ‘equilateral’. For example, a square has sides
of equal length and angles of equal measure. Hence it is a regular polygon. A rectangle is
CHAPTER
3
Understanding
Quadrilaterals
Fig 3.1
Reprint 2024-25
22  MATHEMATICS
equiangular but not equilateral. Is a rectangle a regular polygon? Is an equilateral triangle a
regular polygon? Why?
[Note: Use of  or  indicates segments of equal length].
In the previous classes, have you come across any quadrilateral that is equilateral but not
equiangular? Recall the quadrilateral shapes you saw in earlier classes – Rectangle, Square,
Rhombus etc.
Is there a triangle that is equilateral but not equiangular?
EXERCISE 3.1
1. Given here are some figures.
(1) (2) (3) (4)
(5) (6) (7) (8)
Classify each of them on the basis of the following.
(a) Simple curve (b) Simple closed curve (c) Polygon
(d) Convex polygon (e) Concave polygon
2. What is a regular polygon?
State the name of a regular polygon of
(i) 3 sides (ii) 4 sides (iii) 6 sides
Regular polygons
Polygons that are not regular
Reprint 2024-25
Page 3


UNDERSTANDING QUADRILATERALS  21
3.1  Introduction
Y ou know that the paper is a model for a plane surface. When you join a number of
points without lifting a pencil from the paper (and without retracing any portion of the
drawing other than single points), you get a plane curve.
3.1.1  Convex and concave polygons
A simple closed curve made up of only line segments is called a polygon.
          
Curves that are polygons Curves that are not polygons
Here are some convex polygons and some concave polygons. (Fig 3.1)
Convex polygons Concave polygons
Can you find how these types of polygons differ from one another? Polygons that are
convex have no portions of their diagonals in their exteriors or any line segment joining any
two different points, in the interior of the polygon, lies wholly in the interior of it . Is this
true with concave polygons? Study the figures given. Then try to describe in your own
words what we mean by a convex polygon and what we mean by a concave polygon. Give
two rough sketches of each kind.
In our work in this class, we will be dealing with convex polygons only .
3.1.2  Regular and irregular polygons
A regular polygon is both ‘equiangular’ and ‘equilateral’. For example, a square has sides
of equal length and angles of equal measure. Hence it is a regular polygon. A rectangle is
CHAPTER
3
Understanding
Quadrilaterals
Fig 3.1
Reprint 2024-25
22  MATHEMATICS
equiangular but not equilateral. Is a rectangle a regular polygon? Is an equilateral triangle a
regular polygon? Why?
[Note: Use of  or  indicates segments of equal length].
In the previous classes, have you come across any quadrilateral that is equilateral but not
equiangular? Recall the quadrilateral shapes you saw in earlier classes – Rectangle, Square,
Rhombus etc.
Is there a triangle that is equilateral but not equiangular?
EXERCISE 3.1
1. Given here are some figures.
(1) (2) (3) (4)
(5) (6) (7) (8)
Classify each of them on the basis of the following.
(a) Simple curve (b) Simple closed curve (c) Polygon
(d) Convex polygon (e) Concave polygon
2. What is a regular polygon?
State the name of a regular polygon of
(i) 3 sides (ii) 4 sides (iii) 6 sides
Regular polygons
Polygons that are not regular
Reprint 2024-25
UNDERSTANDING QUADRILATERALS  23
Example 1: Find measure x in Fig 3.3.
Solution: x + 90° + 50° + 110° = 360°      (Why?)
x + 250° = 360°
x = 110°
TRY THESE
Fig 3.4
Fig 3.3
DO THIS
Draw a polygon on the floor, using a piece of chalk.
(In the figure, a pentagon ABCDE is shown) (Fig 3.2).
We want to know the total measure of angles, i.e,
m?1 + m?2 + m?3 + m?4 + m?5. Start at A. Walk along
AB
. On reaching B, you need to turn through an angle of m?1,
to walk along. 
BC
When you reach at C, you need to turn
through an angle of m?2 to walk along 
CD
. Y ou continue to
move in this manner, until you return to side AB. Y ou would
have in fact made one complete turn.
Therefore,  m?1 + m?2 + m?3 + m?4 + m?5  = 360°.
Fig 3.2
This is true whatever be the number of sides of the polygon.
Therefore, the sum of the measures of the external angles of any polygon is 360°.
T ake a regular hexagon Fig 3.4.
1. What is the sum of the measures of its exterior angles x, y, z, p, q, r?
2. Is  x = y = z = p = q = r? Why?
3. What is the measure of each?
(i) exterior angle (ii) interior angle
4. Repeat this activity for the cases of
(i) a regular octagon (ii) a regular 20-gon
Example 2: Find the number of sides of a regular polygon whose each exterior angle
has a measure of 45°.
Solution: Total measure of all exterior angles = 360°
Measure of each exterior angle = 45°
Therefore, the number of exterior angles =  
360
45
= 8
The polygon has 8 sides.
3.2  Sum of the Measures of the Exterior Angles of a
Polygon
On many occasions a knowledge of exterior angles may throw light on the nature of interior
angles and sides.
Reprint 2024-25
Page 4


UNDERSTANDING QUADRILATERALS  21
3.1  Introduction
Y ou know that the paper is a model for a plane surface. When you join a number of
points without lifting a pencil from the paper (and without retracing any portion of the
drawing other than single points), you get a plane curve.
3.1.1  Convex and concave polygons
A simple closed curve made up of only line segments is called a polygon.
          
Curves that are polygons Curves that are not polygons
Here are some convex polygons and some concave polygons. (Fig 3.1)
Convex polygons Concave polygons
Can you find how these types of polygons differ from one another? Polygons that are
convex have no portions of their diagonals in their exteriors or any line segment joining any
two different points, in the interior of the polygon, lies wholly in the interior of it . Is this
true with concave polygons? Study the figures given. Then try to describe in your own
words what we mean by a convex polygon and what we mean by a concave polygon. Give
two rough sketches of each kind.
In our work in this class, we will be dealing with convex polygons only .
3.1.2  Regular and irregular polygons
A regular polygon is both ‘equiangular’ and ‘equilateral’. For example, a square has sides
of equal length and angles of equal measure. Hence it is a regular polygon. A rectangle is
CHAPTER
3
Understanding
Quadrilaterals
Fig 3.1
Reprint 2024-25
22  MATHEMATICS
equiangular but not equilateral. Is a rectangle a regular polygon? Is an equilateral triangle a
regular polygon? Why?
[Note: Use of  or  indicates segments of equal length].
In the previous classes, have you come across any quadrilateral that is equilateral but not
equiangular? Recall the quadrilateral shapes you saw in earlier classes – Rectangle, Square,
Rhombus etc.
Is there a triangle that is equilateral but not equiangular?
EXERCISE 3.1
1. Given here are some figures.
(1) (2) (3) (4)
(5) (6) (7) (8)
Classify each of them on the basis of the following.
(a) Simple curve (b) Simple closed curve (c) Polygon
(d) Convex polygon (e) Concave polygon
2. What is a regular polygon?
State the name of a regular polygon of
(i) 3 sides (ii) 4 sides (iii) 6 sides
Regular polygons
Polygons that are not regular
Reprint 2024-25
UNDERSTANDING QUADRILATERALS  23
Example 1: Find measure x in Fig 3.3.
Solution: x + 90° + 50° + 110° = 360°      (Why?)
x + 250° = 360°
x = 110°
TRY THESE
Fig 3.4
Fig 3.3
DO THIS
Draw a polygon on the floor, using a piece of chalk.
(In the figure, a pentagon ABCDE is shown) (Fig 3.2).
We want to know the total measure of angles, i.e,
m?1 + m?2 + m?3 + m?4 + m?5. Start at A. Walk along
AB
. On reaching B, you need to turn through an angle of m?1,
to walk along. 
BC
When you reach at C, you need to turn
through an angle of m?2 to walk along 
CD
. Y ou continue to
move in this manner, until you return to side AB. Y ou would
have in fact made one complete turn.
Therefore,  m?1 + m?2 + m?3 + m?4 + m?5  = 360°.
Fig 3.2
This is true whatever be the number of sides of the polygon.
Therefore, the sum of the measures of the external angles of any polygon is 360°.
T ake a regular hexagon Fig 3.4.
1. What is the sum of the measures of its exterior angles x, y, z, p, q, r?
2. Is  x = y = z = p = q = r? Why?
3. What is the measure of each?
(i) exterior angle (ii) interior angle
4. Repeat this activity for the cases of
(i) a regular octagon (ii) a regular 20-gon
Example 2: Find the number of sides of a regular polygon whose each exterior angle
has a measure of 45°.
Solution: Total measure of all exterior angles = 360°
Measure of each exterior angle = 45°
Therefore, the number of exterior angles =  
360
45
= 8
The polygon has 8 sides.
3.2  Sum of the Measures of the Exterior Angles of a
Polygon
On many occasions a knowledge of exterior angles may throw light on the nature of interior
angles and sides.
Reprint 2024-25
24  MATHEMATICS
EXERCISE 3.2
1. Find x in the following figures.
(a) (b)
2. Find the measure of each exterior angle of a regular polygon of
(i) 9 sides (ii) 15 sides
3. How many sides does a regular polygon have if the measure of an exterior angle is 24°?
4. How many sides does a regular polygon have if each of its interior angles
is 165°?
5. (a) Is it possible to have a regular polygon with measure of each exterior angle as 22°?
(b) Can it be an interior angle of a regular polygon? Why?
6. (a) What is the minimum interior angle possible for a regular polygon? Why?
(b) What is the maximum exterior angle possible for a regular polygon?
3.3  Kinds of Quadrilaterals
Based on the nature of the sides or angles of a quadrilateral, it gets special names.
3.3.1  Trapezium
Trapezium is a quadrilateral with a pair of parallel sides.
These are trapeziums These are not trapeziums
Study the above figures and discuss with your friends why some of them are trapeziums
while some are not. (Note: The arrow marks indicate parallel lines).
1. T ake identical cut-outs of congruent triangles of sides 3 cm, 4 cm, 5 cm. Arrange
them as shown (Fig 3.5).
Fig 3.5
DO THIS
Reprint 2024-25
Page 5


UNDERSTANDING QUADRILATERALS  21
3.1  Introduction
Y ou know that the paper is a model for a plane surface. When you join a number of
points without lifting a pencil from the paper (and without retracing any portion of the
drawing other than single points), you get a plane curve.
3.1.1  Convex and concave polygons
A simple closed curve made up of only line segments is called a polygon.
          
Curves that are polygons Curves that are not polygons
Here are some convex polygons and some concave polygons. (Fig 3.1)
Convex polygons Concave polygons
Can you find how these types of polygons differ from one another? Polygons that are
convex have no portions of their diagonals in their exteriors or any line segment joining any
two different points, in the interior of the polygon, lies wholly in the interior of it . Is this
true with concave polygons? Study the figures given. Then try to describe in your own
words what we mean by a convex polygon and what we mean by a concave polygon. Give
two rough sketches of each kind.
In our work in this class, we will be dealing with convex polygons only .
3.1.2  Regular and irregular polygons
A regular polygon is both ‘equiangular’ and ‘equilateral’. For example, a square has sides
of equal length and angles of equal measure. Hence it is a regular polygon. A rectangle is
CHAPTER
3
Understanding
Quadrilaterals
Fig 3.1
Reprint 2024-25
22  MATHEMATICS
equiangular but not equilateral. Is a rectangle a regular polygon? Is an equilateral triangle a
regular polygon? Why?
[Note: Use of  or  indicates segments of equal length].
In the previous classes, have you come across any quadrilateral that is equilateral but not
equiangular? Recall the quadrilateral shapes you saw in earlier classes – Rectangle, Square,
Rhombus etc.
Is there a triangle that is equilateral but not equiangular?
EXERCISE 3.1
1. Given here are some figures.
(1) (2) (3) (4)
(5) (6) (7) (8)
Classify each of them on the basis of the following.
(a) Simple curve (b) Simple closed curve (c) Polygon
(d) Convex polygon (e) Concave polygon
2. What is a regular polygon?
State the name of a regular polygon of
(i) 3 sides (ii) 4 sides (iii) 6 sides
Regular polygons
Polygons that are not regular
Reprint 2024-25
UNDERSTANDING QUADRILATERALS  23
Example 1: Find measure x in Fig 3.3.
Solution: x + 90° + 50° + 110° = 360°      (Why?)
x + 250° = 360°
x = 110°
TRY THESE
Fig 3.4
Fig 3.3
DO THIS
Draw a polygon on the floor, using a piece of chalk.
(In the figure, a pentagon ABCDE is shown) (Fig 3.2).
We want to know the total measure of angles, i.e,
m?1 + m?2 + m?3 + m?4 + m?5. Start at A. Walk along
AB
. On reaching B, you need to turn through an angle of m?1,
to walk along. 
BC
When you reach at C, you need to turn
through an angle of m?2 to walk along 
CD
. Y ou continue to
move in this manner, until you return to side AB. Y ou would
have in fact made one complete turn.
Therefore,  m?1 + m?2 + m?3 + m?4 + m?5  = 360°.
Fig 3.2
This is true whatever be the number of sides of the polygon.
Therefore, the sum of the measures of the external angles of any polygon is 360°.
T ake a regular hexagon Fig 3.4.
1. What is the sum of the measures of its exterior angles x, y, z, p, q, r?
2. Is  x = y = z = p = q = r? Why?
3. What is the measure of each?
(i) exterior angle (ii) interior angle
4. Repeat this activity for the cases of
(i) a regular octagon (ii) a regular 20-gon
Example 2: Find the number of sides of a regular polygon whose each exterior angle
has a measure of 45°.
Solution: Total measure of all exterior angles = 360°
Measure of each exterior angle = 45°
Therefore, the number of exterior angles =  
360
45
= 8
The polygon has 8 sides.
3.2  Sum of the Measures of the Exterior Angles of a
Polygon
On many occasions a knowledge of exterior angles may throw light on the nature of interior
angles and sides.
Reprint 2024-25
24  MATHEMATICS
EXERCISE 3.2
1. Find x in the following figures.
(a) (b)
2. Find the measure of each exterior angle of a regular polygon of
(i) 9 sides (ii) 15 sides
3. How many sides does a regular polygon have if the measure of an exterior angle is 24°?
4. How many sides does a regular polygon have if each of its interior angles
is 165°?
5. (a) Is it possible to have a regular polygon with measure of each exterior angle as 22°?
(b) Can it be an interior angle of a regular polygon? Why?
6. (a) What is the minimum interior angle possible for a regular polygon? Why?
(b) What is the maximum exterior angle possible for a regular polygon?
3.3  Kinds of Quadrilaterals
Based on the nature of the sides or angles of a quadrilateral, it gets special names.
3.3.1  Trapezium
Trapezium is a quadrilateral with a pair of parallel sides.
These are trapeziums These are not trapeziums
Study the above figures and discuss with your friends why some of them are trapeziums
while some are not. (Note: The arrow marks indicate parallel lines).
1. T ake identical cut-outs of congruent triangles of sides 3 cm, 4 cm, 5 cm. Arrange
them as shown (Fig 3.5).
Fig 3.5
DO THIS
Reprint 2024-25
UNDERSTANDING QUADRILATERALS  25
DO THIS
You get a trapezium. (Check it!) Which are the parallel sides here? Should the
non-parallel sides be equal?
Y ou can get two more trapeziums using the same set of triangles. Find them out and
discuss their shapes.
2. T ake four set-squares from your and your friend’s instrument boxes. Use different
numbers of them to place side-by-side and obtain different trapeziums.
If the non-parallel sides of a trapezium are of equal length, we call it an isosceles
trapezium. Did you get an isoceles trapezium in any of your investigations given above?
3.3.2  Kite
Kite is a special type of a quadrilateral. The sides with the same markings in each figure
are equal. For example AB = AD and BC = CD.
Fig 3.6
Fig 3.7
Show that
?ABC and
?ADC are
congruent .
What do we
infer from
this?
These are kites These are not kites
Study these figures and try to describe what a kite is. Observe that
(i) A kite has 4 sides (It is a quadrilateral).
(ii) There are exactly two distinct consecutive pairs of sides of equal length.
Check whether a square is a kite.
T ake a thick white sheet.
Fold the paper once.
Draw two line segments of different lengths as shown in Fig 3.6.
Cut along the line segments and open up.
Y ou have the shape of a kite (Fig 3.6).
Has the kite any line symmetry?
Fold both the diagonals of the kite. Use the set-square to check if they cut at
right angles. Are the diagonals equal in length?
V erify (by paper-folding or measurement) if the diagonals bisect each other.
By folding an angle of the kite on its opposite, check for angles of equal measure.
Observe the diagonal folds; do they indicate any diagonal being an angle bisector?
Share your findings with others and list them. A summary of these results are
given elsewhere in the chapter for your reference.
Reprint 2024-25