NCERT Textbook - The Triangle and its Properties

 Page 1


MATHEMATICS 88
6.1 INTRODUCTION
A triangle, you have seen, is a simple closed curve made of three line
segments. It has three vertices, three sides and three angles.
Here is ?ABC (Fig 6.1). It has
Sides: AB , BC , CA
Angles: ?BAC,  ?ABC,  ?BCA
Vertices: A, B, C
The side opposite  to the vertex A is BC. Can you name the angle opposite to the side AB?
Y ou know how to classify triangles based on the (i) sides (ii) angles.
(i) Based on Sides: Scalene, Isosceles and Equilateral triangles.
(ii) Based on Angles: Acute-angled, Obtuse-angled and Right-angled triangles.
Make paper-cut models of the above triangular shapes. Compare your models with
those of your friends and discuss about them.
Fig 6.1
TRY THESE
The Triangle and
its Properties Chapter  6
 1. Write the six elements (i.e., the 3 sides and the 3 angles) of ?ABC.
 2. Write the:
(i) Side opposite to the vertex Q of ?PQR
(ii) Angle opposite to the side LM of ?LMN
(iii) V ertex opposite to the side RT of ?RST
3. Look at Fig 6.2 and classify each of the triangles according to its
(a) Sides
(b) Angles
2024-25
Page 2


MATHEMATICS 88
6.1 INTRODUCTION
A triangle, you have seen, is a simple closed curve made of three line
segments. It has three vertices, three sides and three angles.
Here is ?ABC (Fig 6.1). It has
Sides: AB , BC , CA
Angles: ?BAC,  ?ABC,  ?BCA
Vertices: A, B, C
The side opposite  to the vertex A is BC. Can you name the angle opposite to the side AB?
Y ou know how to classify triangles based on the (i) sides (ii) angles.
(i) Based on Sides: Scalene, Isosceles and Equilateral triangles.
(ii) Based on Angles: Acute-angled, Obtuse-angled and Right-angled triangles.
Make paper-cut models of the above triangular shapes. Compare your models with
those of your friends and discuss about them.
Fig 6.1
TRY THESE
The Triangle and
its Properties Chapter  6
 1. Write the six elements (i.e., the 3 sides and the 3 angles) of ?ABC.
 2. Write the:
(i) Side opposite to the vertex Q of ?PQR
(ii) Angle opposite to the side LM of ?LMN
(iii) V ertex opposite to the side RT of ?RST
3. Look at Fig 6.2 and classify each of the triangles according to its
(a) Sides
(b) Angles
2024-25
THE TRIANGLE AND ITS PROPERTIES 89
Now, let us try to explore something more about triangles.
6.2  MEDIANS OF A TRIANGLE
Given a line segment, you know how to find its perpendicular bisector by paper folding.
Cut out a triangle ABC from a piece of paper (Fig 6.3). Consider any one of its sides, say,
BC . By paper-folding, locate the perpendicular bisector of BC . The folded crease meets
BC at D, its mid-point. Join AD .
Fig 6.3
The line segment AD, joining the mid-point of BC to its opposite vertex A is called a
median of the triangle.
Consider the sides AB and CA and find two more medians of the triangle.
A median connects a vertex of a triangle to the mid-point of the opposite side.
THINK, DISCUSS AND WRITE
1. How many medians can a triangle have?
2. Does a median lie wholly in the interior of the triangle? (If you think that this is not
true, draw a figure to show such a case).
P
Q
R
6cm
10cm
8cm
(ii)
L
M N
7cm
7cm
(iii)
A
B C
D
A
B C
D
Fig 6.2
2024-25
Page 3


MATHEMATICS 88
6.1 INTRODUCTION
A triangle, you have seen, is a simple closed curve made of three line
segments. It has three vertices, three sides and three angles.
Here is ?ABC (Fig 6.1). It has
Sides: AB , BC , CA
Angles: ?BAC,  ?ABC,  ?BCA
Vertices: A, B, C
The side opposite  to the vertex A is BC. Can you name the angle opposite to the side AB?
Y ou know how to classify triangles based on the (i) sides (ii) angles.
(i) Based on Sides: Scalene, Isosceles and Equilateral triangles.
(ii) Based on Angles: Acute-angled, Obtuse-angled and Right-angled triangles.
Make paper-cut models of the above triangular shapes. Compare your models with
those of your friends and discuss about them.
Fig 6.1
TRY THESE
The Triangle and
its Properties Chapter  6
 1. Write the six elements (i.e., the 3 sides and the 3 angles) of ?ABC.
 2. Write the:
(i) Side opposite to the vertex Q of ?PQR
(ii) Angle opposite to the side LM of ?LMN
(iii) V ertex opposite to the side RT of ?RST
3. Look at Fig 6.2 and classify each of the triangles according to its
(a) Sides
(b) Angles
2024-25
THE TRIANGLE AND ITS PROPERTIES 89
Now, let us try to explore something more about triangles.
6.2  MEDIANS OF A TRIANGLE
Given a line segment, you know how to find its perpendicular bisector by paper folding.
Cut out a triangle ABC from a piece of paper (Fig 6.3). Consider any one of its sides, say,
BC . By paper-folding, locate the perpendicular bisector of BC . The folded crease meets
BC at D, its mid-point. Join AD .
Fig 6.3
The line segment AD, joining the mid-point of BC to its opposite vertex A is called a
median of the triangle.
Consider the sides AB and CA and find two more medians of the triangle.
A median connects a vertex of a triangle to the mid-point of the opposite side.
THINK, DISCUSS AND WRITE
1. How many medians can a triangle have?
2. Does a median lie wholly in the interior of the triangle? (If you think that this is not
true, draw a figure to show such a case).
P
Q
R
6cm
10cm
8cm
(ii)
L
M N
7cm
7cm
(iii)
A
B C
D
A
B C
D
Fig 6.2
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MATHEMATICS 90
6.3  ALTITUDES OF A TRIANGLE
Make a triangular shaped cardboard ABC. Place it
upright on a table. How ‘tall’ is the triangle? The height
is the distance from vertex A (in the Fig 6.4) to the
base BC .
From A to BC , you can think of many line
segments (see the next Fig 6.5). Which among them
will represent its height?
The height is given by the line segment that starts from A,
comes straight down to BC , and is perpendicular to BC .
This line segment AL is an altitude of the triangle.
An altitude has one end point at a vertex of the triangle
and the other on the line containing the opposite side. Through
each vertex, an altitude can be drawn.
THINK, DISCUSS AND WRITE
1. How many altitudes can a triangle have?
2. Draw rough sketches of altitudes from A to BC for the following triangles (Fig 6.6):
A
B C
L
Fig 6.5
A
B C
A
B C
A
B
C
DO THIS
A
B C
Fig 6.4
Acute-angled Right-angled Obtuse-angled
(i) (ii) (iii)
Fig 6.6
3. W ill an altitude always lie in the interior of a triangle? If you think that this need not be
true, draw a rough sketch to show such a case.
4. Can you think of a triangle in which two altitudes of the triangle are two of its sides?
5. Can the altitude and median be same for a triangle?
(Hint: For Q.No.  4 and 5, investigate by drawing the altitudes for every type of triangle).
Take several cut-outs of
 (i) an equilateral triangle (ii) an isosceles triangle and
(iii) a scalene triangle.
Find their altitudes and medians. Do you find anything special about them? Discuss it
with your friends.
2024-25
Page 4


MATHEMATICS 88
6.1 INTRODUCTION
A triangle, you have seen, is a simple closed curve made of three line
segments. It has three vertices, three sides and three angles.
Here is ?ABC (Fig 6.1). It has
Sides: AB , BC , CA
Angles: ?BAC,  ?ABC,  ?BCA
Vertices: A, B, C
The side opposite  to the vertex A is BC. Can you name the angle opposite to the side AB?
Y ou know how to classify triangles based on the (i) sides (ii) angles.
(i) Based on Sides: Scalene, Isosceles and Equilateral triangles.
(ii) Based on Angles: Acute-angled, Obtuse-angled and Right-angled triangles.
Make paper-cut models of the above triangular shapes. Compare your models with
those of your friends and discuss about them.
Fig 6.1
TRY THESE
The Triangle and
its Properties Chapter  6
 1. Write the six elements (i.e., the 3 sides and the 3 angles) of ?ABC.
 2. Write the:
(i) Side opposite to the vertex Q of ?PQR
(ii) Angle opposite to the side LM of ?LMN
(iii) V ertex opposite to the side RT of ?RST
3. Look at Fig 6.2 and classify each of the triangles according to its
(a) Sides
(b) Angles
2024-25
THE TRIANGLE AND ITS PROPERTIES 89
Now, let us try to explore something more about triangles.
6.2  MEDIANS OF A TRIANGLE
Given a line segment, you know how to find its perpendicular bisector by paper folding.
Cut out a triangle ABC from a piece of paper (Fig 6.3). Consider any one of its sides, say,
BC . By paper-folding, locate the perpendicular bisector of BC . The folded crease meets
BC at D, its mid-point. Join AD .
Fig 6.3
The line segment AD, joining the mid-point of BC to its opposite vertex A is called a
median of the triangle.
Consider the sides AB and CA and find two more medians of the triangle.
A median connects a vertex of a triangle to the mid-point of the opposite side.
THINK, DISCUSS AND WRITE
1. How many medians can a triangle have?
2. Does a median lie wholly in the interior of the triangle? (If you think that this is not
true, draw a figure to show such a case).
P
Q
R
6cm
10cm
8cm
(ii)
L
M N
7cm
7cm
(iii)
A
B C
D
A
B C
D
Fig 6.2
2024-25
MATHEMATICS 90
6.3  ALTITUDES OF A TRIANGLE
Make a triangular shaped cardboard ABC. Place it
upright on a table. How ‘tall’ is the triangle? The height
is the distance from vertex A (in the Fig 6.4) to the
base BC .
From A to BC , you can think of many line
segments (see the next Fig 6.5). Which among them
will represent its height?
The height is given by the line segment that starts from A,
comes straight down to BC , and is perpendicular to BC .
This line segment AL is an altitude of the triangle.
An altitude has one end point at a vertex of the triangle
and the other on the line containing the opposite side. Through
each vertex, an altitude can be drawn.
THINK, DISCUSS AND WRITE
1. How many altitudes can a triangle have?
2. Draw rough sketches of altitudes from A to BC for the following triangles (Fig 6.6):
A
B C
L
Fig 6.5
A
B C
A
B C
A
B
C
DO THIS
A
B C
Fig 6.4
Acute-angled Right-angled Obtuse-angled
(i) (ii) (iii)
Fig 6.6
3. W ill an altitude always lie in the interior of a triangle? If you think that this need not be
true, draw a rough sketch to show such a case.
4. Can you think of a triangle in which two altitudes of the triangle are two of its sides?
5. Can the altitude and median be same for a triangle?
(Hint: For Q.No.  4 and 5, investigate by drawing the altitudes for every type of triangle).
Take several cut-outs of
 (i) an equilateral triangle (ii) an isosceles triangle and
(iii) a scalene triangle.
Find their altitudes and medians. Do you find anything special about them? Discuss it
with your friends.
2024-25
THE TRIANGLE AND ITS PROPERTIES 91
EXERCISE 6.1
1. In ? PQR, D is the mid-point of QR .
PM is _________________.
PD
 is _________________.
Is QM = MR?
2. Draw rough sketches for the following:
(a) In ?ABC, BE is a median.
(b) In ?PQR, PQ and PR are altitudes of the triangle.
(c) In ?XYZ, YL is an altitude in the exterior of the triangle.
3. V erify by drawing a diagram if the median and altitude of an isosceles triangle can be
same.
P
Q R
D M
DO THIS
Fig 6.7
Fig 6.8
6.4 EXTERIOR ANGLE OF A TRIANGLE AND ITS PROPERTY
1. Draw a triangle ABC and produce one of its sides,
say BC as shown in Fig 6.7. Observe the angle ACD
formed at the point C. This angle lies in the exterior
of ?ABC. W e call it an exterior angle of the ?ABC
formed at vertex C.
Clearly ? BCA is an adjacent angle to ?ACD. The
remaining two angles of the triangle namely ? A and
?B are called  the two interior opposite angles or the two remote interior angles
of  ?ACD. Now cut out (or make trace copies of) ?A and ?B and place them
adjacent to each other as shown in Fig 6.8.
Do these two pieces together entirely cover ?ACD?
Can you say that
m ?ACD = m ?A + m ?B?
2. As done earlier, draw a triangle ABC and form an
exterior angle ACD. Now take a protractor and
measure ?ACD, ?A and ?B.
Find the sum ?A + ?B and compare it with the
measure of ?ACD. Do you observe that ?ACD is
equal (or nearly equal, if there is an error in
measurement) to ?A + ?B?
2024-25
Page 5


MATHEMATICS 88
6.1 INTRODUCTION
A triangle, you have seen, is a simple closed curve made of three line
segments. It has three vertices, three sides and three angles.
Here is ?ABC (Fig 6.1). It has
Sides: AB , BC , CA
Angles: ?BAC,  ?ABC,  ?BCA
Vertices: A, B, C
The side opposite  to the vertex A is BC. Can you name the angle opposite to the side AB?
Y ou know how to classify triangles based on the (i) sides (ii) angles.
(i) Based on Sides: Scalene, Isosceles and Equilateral triangles.
(ii) Based on Angles: Acute-angled, Obtuse-angled and Right-angled triangles.
Make paper-cut models of the above triangular shapes. Compare your models with
those of your friends and discuss about them.
Fig 6.1
TRY THESE
The Triangle and
its Properties Chapter  6
 1. Write the six elements (i.e., the 3 sides and the 3 angles) of ?ABC.
 2. Write the:
(i) Side opposite to the vertex Q of ?PQR
(ii) Angle opposite to the side LM of ?LMN
(iii) V ertex opposite to the side RT of ?RST
3. Look at Fig 6.2 and classify each of the triangles according to its
(a) Sides
(b) Angles
2024-25
THE TRIANGLE AND ITS PROPERTIES 89
Now, let us try to explore something more about triangles.
6.2  MEDIANS OF A TRIANGLE
Given a line segment, you know how to find its perpendicular bisector by paper folding.
Cut out a triangle ABC from a piece of paper (Fig 6.3). Consider any one of its sides, say,
BC . By paper-folding, locate the perpendicular bisector of BC . The folded crease meets
BC at D, its mid-point. Join AD .
Fig 6.3
The line segment AD, joining the mid-point of BC to its opposite vertex A is called a
median of the triangle.
Consider the sides AB and CA and find two more medians of the triangle.
A median connects a vertex of a triangle to the mid-point of the opposite side.
THINK, DISCUSS AND WRITE
1. How many medians can a triangle have?
2. Does a median lie wholly in the interior of the triangle? (If you think that this is not
true, draw a figure to show such a case).
P
Q
R
6cm
10cm
8cm
(ii)
L
M N
7cm
7cm
(iii)
A
B C
D
A
B C
D
Fig 6.2
2024-25
MATHEMATICS 90
6.3  ALTITUDES OF A TRIANGLE
Make a triangular shaped cardboard ABC. Place it
upright on a table. How ‘tall’ is the triangle? The height
is the distance from vertex A (in the Fig 6.4) to the
base BC .
From A to BC , you can think of many line
segments (see the next Fig 6.5). Which among them
will represent its height?
The height is given by the line segment that starts from A,
comes straight down to BC , and is perpendicular to BC .
This line segment AL is an altitude of the triangle.
An altitude has one end point at a vertex of the triangle
and the other on the line containing the opposite side. Through
each vertex, an altitude can be drawn.
THINK, DISCUSS AND WRITE
1. How many altitudes can a triangle have?
2. Draw rough sketches of altitudes from A to BC for the following triangles (Fig 6.6):
A
B C
L
Fig 6.5
A
B C
A
B C
A
B
C
DO THIS
A
B C
Fig 6.4
Acute-angled Right-angled Obtuse-angled
(i) (ii) (iii)
Fig 6.6
3. W ill an altitude always lie in the interior of a triangle? If you think that this need not be
true, draw a rough sketch to show such a case.
4. Can you think of a triangle in which two altitudes of the triangle are two of its sides?
5. Can the altitude and median be same for a triangle?
(Hint: For Q.No.  4 and 5, investigate by drawing the altitudes for every type of triangle).
Take several cut-outs of
 (i) an equilateral triangle (ii) an isosceles triangle and
(iii) a scalene triangle.
Find their altitudes and medians. Do you find anything special about them? Discuss it
with your friends.
2024-25
THE TRIANGLE AND ITS PROPERTIES 91
EXERCISE 6.1
1. In ? PQR, D is the mid-point of QR .
PM is _________________.
PD
 is _________________.
Is QM = MR?
2. Draw rough sketches for the following:
(a) In ?ABC, BE is a median.
(b) In ?PQR, PQ and PR are altitudes of the triangle.
(c) In ?XYZ, YL is an altitude in the exterior of the triangle.
3. V erify by drawing a diagram if the median and altitude of an isosceles triangle can be
same.
P
Q R
D M
DO THIS
Fig 6.7
Fig 6.8
6.4 EXTERIOR ANGLE OF A TRIANGLE AND ITS PROPERTY
1. Draw a triangle ABC and produce one of its sides,
say BC as shown in Fig 6.7. Observe the angle ACD
formed at the point C. This angle lies in the exterior
of ?ABC. W e call it an exterior angle of the ?ABC
formed at vertex C.
Clearly ? BCA is an adjacent angle to ?ACD. The
remaining two angles of the triangle namely ? A and
?B are called  the two interior opposite angles or the two remote interior angles
of  ?ACD. Now cut out (or make trace copies of) ?A and ?B and place them
adjacent to each other as shown in Fig 6.8.
Do these two pieces together entirely cover ?ACD?
Can you say that
m ?ACD = m ?A + m ?B?
2. As done earlier, draw a triangle ABC and form an
exterior angle ACD. Now take a protractor and
measure ?ACD, ?A and ?B.
Find the sum ?A + ?B and compare it with the
measure of ?ACD. Do you observe that ?ACD is
equal (or nearly equal, if there is an error in
measurement) to ?A + ?B?
2024-25
MATHEMATICS 92
Y ou may repeat the two activities as mentioned by drawing some more triangles along
with their exterior angles. Every time, you will find that the exterior angle of a triangle is
equal to the sum of its two interior opposite angles.
A logical step-by-step argument can further confirm this fact.
An exterior angle of a triangle is equal to the sum of its interior opposite
angles.
Given: Consider ?ABC.
?ACD is an exterior angle.
To Show: m?ACD =  m?A + m?B
Through C draw CE , parallel to BA .
Justification
Steps Reasons
(a) ?1 = ?x BA CE || and AC is a transversal.
Therefore, alternate angles should be equal.
(b) ?2 = ?y
BA CE ||
 and 
BD
 is a transversal.
Therefore, corresponding angles should be equal.
(c) ?1 + ?2 = ?x + ?y
(d) Now,  ?x + ?y = m ?ACD From Fig 6.9
Hence, ?1 + ?2 = ?ACD
The above relation between an exterior angle and its two interior opposite angles is
referred to as the Exterior Angle Property of a triangle.
THINK, DISCUSS AND WRITE
1. Exterior angles can be formed for a triangle in many ways. Three of them are shown
here (Fig 6.10)
Fig 6.10
There are three more ways of getting exterior angles. Try to produce those rough
sketches.
2. Are the exterior angles formed at each vertex of a triangle equal?
3. What can you say about the sum of an exterior angle of a triangle and its adjacent
interior angle?
Fig 6.9
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