NCERT Textbook: Triangles | Mathematics (Maths) Class 9 PDF Download

 Page 1


CHAPTER 7
TRIANGLES
7.1 Introduction
You have studied about triangles and their various properties in your earlier classes.
You know that a closed figure formed by three intersecting lines is called a triangle.
(‘Tri’ means ‘three’). A triangle has three sides, three angles and three vertices. For
example, in triangle ABC, denoted as ? ABC (see Fig. 7.1); AB, BC, CA are the three
sides, ? A, ? B, ? C are the three angles and A, B, C are three vertices.
In Chapter 6, you have also studied some properties
of triangles. In this chapter, you will study in details
about the congruence of triangles, rules of congruence,
some more properties of triangles and inequalities in
a triangle. You have already verified most of these
properties in earlier classes. We will now prove some
of them.
7.2 Congruence of Triangles
You must have observed that two copies of your photographs of the same size are
identical. Similarly, two bangles of the same size, two A TM cards issued by the same
bank are identical. You may recall that on placing a one rupee coin on another minted
in the same year, they cover each other completely.
Do you remember what such figures are called? Indeed they are called congruent
figures (‘congruent’ means equal in all respects or figures whose shapes and sizes
are both the same).
Now, draw two circles of the same radius and place one on the other. What do
you observe? They cover each other completely and we call them as congruent circles.
Fig. 7.1
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Page 2


CHAPTER 7
TRIANGLES
7.1 Introduction
You have studied about triangles and their various properties in your earlier classes.
You know that a closed figure formed by three intersecting lines is called a triangle.
(‘Tri’ means ‘three’). A triangle has three sides, three angles and three vertices. For
example, in triangle ABC, denoted as ? ABC (see Fig. 7.1); AB, BC, CA are the three
sides, ? A, ? B, ? C are the three angles and A, B, C are three vertices.
In Chapter 6, you have also studied some properties
of triangles. In this chapter, you will study in details
about the congruence of triangles, rules of congruence,
some more properties of triangles and inequalities in
a triangle. You have already verified most of these
properties in earlier classes. We will now prove some
of them.
7.2 Congruence of Triangles
You must have observed that two copies of your photographs of the same size are
identical. Similarly, two bangles of the same size, two A TM cards issued by the same
bank are identical. You may recall that on placing a one rupee coin on another minted
in the same year, they cover each other completely.
Do you remember what such figures are called? Indeed they are called congruent
figures (‘congruent’ means equal in all respects or figures whose shapes and sizes
are both the same).
Now, draw two circles of the same radius and place one on the other. What do
you observe? They cover each other completely and we call them as congruent circles.
Fig. 7.1
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84 MATHEMA TICS
Repeat this activity by placing one
square on the other with sides of the same
measure (see Fig. 7.2) or by placing two
equilateral triangles of equal sides on each
other. You will observe that the squares are
congruent to each other and so are the
equilateral triangles.
You may wonder why we are studying congruence. Y ou all must have seen the ice
tray in your refrigerator. Observe that the moulds for making ice are all congruent.
The cast used for moulding in the tray also has congruent depressions (may be all are
rectangular or all circular or all triangular). So, whenever identical objects have to be
produced, the concept of congruence is used in making the cast.
Sometimes, you may find it difficult to replace the refill in your pen by a new one
and this is so when the new refill is not of the same size as the one you want to
remove. Obviously, if the two refills are identical or congruent, the new refill fits.
So, you can find numerous examples where congruence of objects is applied in
daily life situations.
Can you think of some more examples of congruent figures?
Now, which of the following figures are not congruent to the square in
Fig 7.3 (i) :
Fig. 7.3
The large squares in Fig. 7.3 (ii) and (iii) are obviously not congruent to the one in
Fig 7.3 (i), but the square in Fig 7.3 (iv) is congruent to the one given in Fig 7.3 (i).
Let us now discuss the congruence of two triangles.
You already know that two triangles are congruent if the sides and angles of one
triangle are equal to the corresponding sides and angles of the other triangle.
Fig. 7.2
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Page 3


CHAPTER 7
TRIANGLES
7.1 Introduction
You have studied about triangles and their various properties in your earlier classes.
You know that a closed figure formed by three intersecting lines is called a triangle.
(‘Tri’ means ‘three’). A triangle has three sides, three angles and three vertices. For
example, in triangle ABC, denoted as ? ABC (see Fig. 7.1); AB, BC, CA are the three
sides, ? A, ? B, ? C are the three angles and A, B, C are three vertices.
In Chapter 6, you have also studied some properties
of triangles. In this chapter, you will study in details
about the congruence of triangles, rules of congruence,
some more properties of triangles and inequalities in
a triangle. You have already verified most of these
properties in earlier classes. We will now prove some
of them.
7.2 Congruence of Triangles
You must have observed that two copies of your photographs of the same size are
identical. Similarly, two bangles of the same size, two A TM cards issued by the same
bank are identical. You may recall that on placing a one rupee coin on another minted
in the same year, they cover each other completely.
Do you remember what such figures are called? Indeed they are called congruent
figures (‘congruent’ means equal in all respects or figures whose shapes and sizes
are both the same).
Now, draw two circles of the same radius and place one on the other. What do
you observe? They cover each other completely and we call them as congruent circles.
Fig. 7.1
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84 MATHEMA TICS
Repeat this activity by placing one
square on the other with sides of the same
measure (see Fig. 7.2) or by placing two
equilateral triangles of equal sides on each
other. You will observe that the squares are
congruent to each other and so are the
equilateral triangles.
You may wonder why we are studying congruence. Y ou all must have seen the ice
tray in your refrigerator. Observe that the moulds for making ice are all congruent.
The cast used for moulding in the tray also has congruent depressions (may be all are
rectangular or all circular or all triangular). So, whenever identical objects have to be
produced, the concept of congruence is used in making the cast.
Sometimes, you may find it difficult to replace the refill in your pen by a new one
and this is so when the new refill is not of the same size as the one you want to
remove. Obviously, if the two refills are identical or congruent, the new refill fits.
So, you can find numerous examples where congruence of objects is applied in
daily life situations.
Can you think of some more examples of congruent figures?
Now, which of the following figures are not congruent to the square in
Fig 7.3 (i) :
Fig. 7.3
The large squares in Fig. 7.3 (ii) and (iii) are obviously not congruent to the one in
Fig 7.3 (i), but the square in Fig 7.3 (iv) is congruent to the one given in Fig 7.3 (i).
Let us now discuss the congruence of two triangles.
You already know that two triangles are congruent if the sides and angles of one
triangle are equal to the corresponding sides and angles of the other triangle.
Fig. 7.2
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TRIANGLES 85
Now, which of the triangles given below are congruent to triangle ABC in
Fig. 7.4 (i)?
Fig. 7.4
Cut out each of these triangles from Fig. 7.4 (ii) to (v) and turn them around and
try to cover ? ABC. Observe that triangles in Fig. 7.4 (ii), (iii) and (iv) are congruent
to ? ABC while ? TSU of Fig 7.4 (v) is not congruent to ? ABC.
If ? PQR is congruent to ? ABC, we write ? PQR ? ? ABC.
Notice that when ? PQR ? ? ABC, then sides of ? PQR fall on corresponding
equal sides of ? ABC and so is the case for the angles.
That is, PQ covers AB, QR covers BC and RP covers CA; ? P  covers ? A,
? Q covers ? B and ? R covers ? C. Also, there is a one-one correspondence
between the vertices. That is, P corresponds to A, Q to B, R to C and so on which is
written as
P ? A, Q ? B, R ? C
Note that under this correspondence, ? PQR ? ? ABC; but it will not be correct to
write ?QRP ? ? ABC.
Similarly , for Fig. 7.4 (iii),
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Page 4


CHAPTER 7
TRIANGLES
7.1 Introduction
You have studied about triangles and their various properties in your earlier classes.
You know that a closed figure formed by three intersecting lines is called a triangle.
(‘Tri’ means ‘three’). A triangle has three sides, three angles and three vertices. For
example, in triangle ABC, denoted as ? ABC (see Fig. 7.1); AB, BC, CA are the three
sides, ? A, ? B, ? C are the three angles and A, B, C are three vertices.
In Chapter 6, you have also studied some properties
of triangles. In this chapter, you will study in details
about the congruence of triangles, rules of congruence,
some more properties of triangles and inequalities in
a triangle. You have already verified most of these
properties in earlier classes. We will now prove some
of them.
7.2 Congruence of Triangles
You must have observed that two copies of your photographs of the same size are
identical. Similarly, two bangles of the same size, two A TM cards issued by the same
bank are identical. You may recall that on placing a one rupee coin on another minted
in the same year, they cover each other completely.
Do you remember what such figures are called? Indeed they are called congruent
figures (‘congruent’ means equal in all respects or figures whose shapes and sizes
are both the same).
Now, draw two circles of the same radius and place one on the other. What do
you observe? They cover each other completely and we call them as congruent circles.
Fig. 7.1
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84 MATHEMA TICS
Repeat this activity by placing one
square on the other with sides of the same
measure (see Fig. 7.2) or by placing two
equilateral triangles of equal sides on each
other. You will observe that the squares are
congruent to each other and so are the
equilateral triangles.
You may wonder why we are studying congruence. Y ou all must have seen the ice
tray in your refrigerator. Observe that the moulds for making ice are all congruent.
The cast used for moulding in the tray also has congruent depressions (may be all are
rectangular or all circular or all triangular). So, whenever identical objects have to be
produced, the concept of congruence is used in making the cast.
Sometimes, you may find it difficult to replace the refill in your pen by a new one
and this is so when the new refill is not of the same size as the one you want to
remove. Obviously, if the two refills are identical or congruent, the new refill fits.
So, you can find numerous examples where congruence of objects is applied in
daily life situations.
Can you think of some more examples of congruent figures?
Now, which of the following figures are not congruent to the square in
Fig 7.3 (i) :
Fig. 7.3
The large squares in Fig. 7.3 (ii) and (iii) are obviously not congruent to the one in
Fig 7.3 (i), but the square in Fig 7.3 (iv) is congruent to the one given in Fig 7.3 (i).
Let us now discuss the congruence of two triangles.
You already know that two triangles are congruent if the sides and angles of one
triangle are equal to the corresponding sides and angles of the other triangle.
Fig. 7.2
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TRIANGLES 85
Now, which of the triangles given below are congruent to triangle ABC in
Fig. 7.4 (i)?
Fig. 7.4
Cut out each of these triangles from Fig. 7.4 (ii) to (v) and turn them around and
try to cover ? ABC. Observe that triangles in Fig. 7.4 (ii), (iii) and (iv) are congruent
to ? ABC while ? TSU of Fig 7.4 (v) is not congruent to ? ABC.
If ? PQR is congruent to ? ABC, we write ? PQR ? ? ABC.
Notice that when ? PQR ? ? ABC, then sides of ? PQR fall on corresponding
equal sides of ? ABC and so is the case for the angles.
That is, PQ covers AB, QR covers BC and RP covers CA; ? P  covers ? A,
? Q covers ? B and ? R covers ? C. Also, there is a one-one correspondence
between the vertices. That is, P corresponds to A, Q to B, R to C and so on which is
written as
P ? A, Q ? B, R ? C
Note that under this correspondence, ? PQR ? ? ABC; but it will not be correct to
write ?QRP ? ? ABC.
Similarly , for Fig. 7.4 (iii),
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86 MATHEMA TICS
FD ? AB, DE ? BC and EF ? CA
and F ? A, D ? B and E ? C
So, ? FDE ? ? ABC but writing ? DEF ? ? ABC is not correct.
Give the correspondence between the triangle in Fig. 7.4 (iv) and ? ABC.
So, it is necessary to write the correspondence of vertices correctly for writing of
congruence of triangles in symbolic form.
Note that in congruent triangles corresponding parts are equal and we write
in short ‘CPCT’ for corresponding parts of congruent triangles.
7.3 Criteria for Congruence of Triangles
In earlier classes, you have learnt four criteria for congruence of triangles. Let us
recall them.
Draw two triangles with one side 3 cm. Are these triangles congruent? Observe
that they are not congruent (see Fig. 7.5).
Fig. 7.5
Now, draw two triangles with one side 4 cm and one angle 50° (see Fig. 7.6). Are
they congruent?
Fig. 7.6
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Page 5


CHAPTER 7
TRIANGLES
7.1 Introduction
You have studied about triangles and their various properties in your earlier classes.
You know that a closed figure formed by three intersecting lines is called a triangle.
(‘Tri’ means ‘three’). A triangle has three sides, three angles and three vertices. For
example, in triangle ABC, denoted as ? ABC (see Fig. 7.1); AB, BC, CA are the three
sides, ? A, ? B, ? C are the three angles and A, B, C are three vertices.
In Chapter 6, you have also studied some properties
of triangles. In this chapter, you will study in details
about the congruence of triangles, rules of congruence,
some more properties of triangles and inequalities in
a triangle. You have already verified most of these
properties in earlier classes. We will now prove some
of them.
7.2 Congruence of Triangles
You must have observed that two copies of your photographs of the same size are
identical. Similarly, two bangles of the same size, two A TM cards issued by the same
bank are identical. You may recall that on placing a one rupee coin on another minted
in the same year, they cover each other completely.
Do you remember what such figures are called? Indeed they are called congruent
figures (‘congruent’ means equal in all respects or figures whose shapes and sizes
are both the same).
Now, draw two circles of the same radius and place one on the other. What do
you observe? They cover each other completely and we call them as congruent circles.
Fig. 7.1
2024-25
84 MATHEMA TICS
Repeat this activity by placing one
square on the other with sides of the same
measure (see Fig. 7.2) or by placing two
equilateral triangles of equal sides on each
other. You will observe that the squares are
congruent to each other and so are the
equilateral triangles.
You may wonder why we are studying congruence. Y ou all must have seen the ice
tray in your refrigerator. Observe that the moulds for making ice are all congruent.
The cast used for moulding in the tray also has congruent depressions (may be all are
rectangular or all circular or all triangular). So, whenever identical objects have to be
produced, the concept of congruence is used in making the cast.
Sometimes, you may find it difficult to replace the refill in your pen by a new one
and this is so when the new refill is not of the same size as the one you want to
remove. Obviously, if the two refills are identical or congruent, the new refill fits.
So, you can find numerous examples where congruence of objects is applied in
daily life situations.
Can you think of some more examples of congruent figures?
Now, which of the following figures are not congruent to the square in
Fig 7.3 (i) :
Fig. 7.3
The large squares in Fig. 7.3 (ii) and (iii) are obviously not congruent to the one in
Fig 7.3 (i), but the square in Fig 7.3 (iv) is congruent to the one given in Fig 7.3 (i).
Let us now discuss the congruence of two triangles.
You already know that two triangles are congruent if the sides and angles of one
triangle are equal to the corresponding sides and angles of the other triangle.
Fig. 7.2
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TRIANGLES 85
Now, which of the triangles given below are congruent to triangle ABC in
Fig. 7.4 (i)?
Fig. 7.4
Cut out each of these triangles from Fig. 7.4 (ii) to (v) and turn them around and
try to cover ? ABC. Observe that triangles in Fig. 7.4 (ii), (iii) and (iv) are congruent
to ? ABC while ? TSU of Fig 7.4 (v) is not congruent to ? ABC.
If ? PQR is congruent to ? ABC, we write ? PQR ? ? ABC.
Notice that when ? PQR ? ? ABC, then sides of ? PQR fall on corresponding
equal sides of ? ABC and so is the case for the angles.
That is, PQ covers AB, QR covers BC and RP covers CA; ? P  covers ? A,
? Q covers ? B and ? R covers ? C. Also, there is a one-one correspondence
between the vertices. That is, P corresponds to A, Q to B, R to C and so on which is
written as
P ? A, Q ? B, R ? C
Note that under this correspondence, ? PQR ? ? ABC; but it will not be correct to
write ?QRP ? ? ABC.
Similarly , for Fig. 7.4 (iii),
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86 MATHEMA TICS
FD ? AB, DE ? BC and EF ? CA
and F ? A, D ? B and E ? C
So, ? FDE ? ? ABC but writing ? DEF ? ? ABC is not correct.
Give the correspondence between the triangle in Fig. 7.4 (iv) and ? ABC.
So, it is necessary to write the correspondence of vertices correctly for writing of
congruence of triangles in symbolic form.
Note that in congruent triangles corresponding parts are equal and we write
in short ‘CPCT’ for corresponding parts of congruent triangles.
7.3 Criteria for Congruence of Triangles
In earlier classes, you have learnt four criteria for congruence of triangles. Let us
recall them.
Draw two triangles with one side 3 cm. Are these triangles congruent? Observe
that they are not congruent (see Fig. 7.5).
Fig. 7.5
Now, draw two triangles with one side 4 cm and one angle 50° (see Fig. 7.6). Are
they congruent?
Fig. 7.6
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TRIANGLES 87
See that these two triangles are not congruent.
Repeat this activity with some more pairs of triangles.
So, equality of one pair of sides or one pair of sides and one pair of angles is not
sufficient to give us congruent triangles.
What would happen if the other pair of arms (sides) of the equal angles are also
equal?
In Fig 7.7, BC = QR, ? B = ? Q and also, AB = PQ. Now, what can you say
about congruence of ? ABC and ? PQR?
Recall from your earlier classes that, in this case, the two triangles are congruent.
Verify this for ? ABC and ? PQR in Fig. 7.7.
Repeat this activity with other pairs of triangles. Do you observe that the equality
of two sides and the included angle is enough for the congruence of triangles? Yes, it
is enough.
Fig. 7.7
This is the first criterion for congruence of triangles.
Axiom 7.1 (SAS congruence rule) : Two triangles are congruent if two sides
and the included angle of one triangle are equal to the two sides and the included
angle of the other triangle.
This result cannot be proved with the help of previously known results and so it is
accepted true as an axiom (see Appendix 1).
Let us now take some examples.
Example 1 : In Fig. 7.8, OA = OB and OD = OC. Show that
(i) ? AOD ? ? BOC and (ii) AD || BC.
Solution : (i) You may observe that in ? AOD and ? BOC,
OA = OB
(Given)
OD = OC
Fig. 7.8
?
?
?
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