NCERT Textbook: Congruence of Triangles | NCERT Textbooks (Class 6 to Class 12) - CTET & State TET PDF Download

 Page 1


CONGRUENCE OF TRIANGLES 133 133 133 133 133
7.1  INTRODUCTION
Y ou are now ready to learn a very important geometrical idea, Congruence. In particular,
you will study a lot about congruence of triangles.
T o understand what congruence is, we turn to some activities.
T ake two stamps (Fig 7.1) of same denomination. Place one stamp over
the other. What do you observe?
Fig 7.1
One stamp covers the other completely and exactly. This means that the two stamps are
of the same shape and same size. Such objects are said to be congruent. The two stamps
used by you are congruent to one another. Congruent objects are exact copies of one
another.
Can you, now, say if the following objects are congruent or not?
 1. Shaving blades of the same company [Fig 7.2 (i)].
 2. Sheets of the same letter-pad [Fig 7.2 (ii)]. 3. Biscuits in the same packet [Fig 7.2 (iii)].
4. T oys made of the same mould. [Fig 7.2(iv)]
(i) (ii) (iii) (iv)
Chapter  7
Congruence of
Triangles
DO THIS
Fig 7.2
2022-23
Page 2


CONGRUENCE OF TRIANGLES 133 133 133 133 133
7.1  INTRODUCTION
Y ou are now ready to learn a very important geometrical idea, Congruence. In particular,
you will study a lot about congruence of triangles.
T o understand what congruence is, we turn to some activities.
T ake two stamps (Fig 7.1) of same denomination. Place one stamp over
the other. What do you observe?
Fig 7.1
One stamp covers the other completely and exactly. This means that the two stamps are
of the same shape and same size. Such objects are said to be congruent. The two stamps
used by you are congruent to one another. Congruent objects are exact copies of one
another.
Can you, now, say if the following objects are congruent or not?
 1. Shaving blades of the same company [Fig 7.2 (i)].
 2. Sheets of the same letter-pad [Fig 7.2 (ii)]. 3. Biscuits in the same packet [Fig 7.2 (iii)].
4. T oys made of the same mould. [Fig 7.2(iv)]
(i) (ii) (iii) (iv)
Chapter  7
Congruence of
Triangles
DO THIS
Fig 7.2
2022-23
MATHEMATICS 134 134 134 134 134
The relation of two objects being congruent is called congruence. For the present,
we will deal with plane figures only, although congruence is a general idea applicable to
three-dimensional shapes also. W e will try to learn a precise meaning of the congruence
of plane figures already known.
7.2  CONGRUENCE OF PLANE FIGURES
Look at the two figures given here (Fig 7.3). Are they congruent?
(i) (ii)
Fig 7.3
Y ou can use the method of superposition. T ake a trace-copy of one of them and place
it over the other . If the figures cover each other completely , they are congruent. Alternatively ,
you may cut out one of them and place it over the other. Beware! Y ou are not allowed to
bend, twist or stretch the figure that is cut out (or traced out).
In Fig 7.3, if figure F
1
 is congruent to figure F
2 
, we write F
1
 ? F
2
.
7.3  CONGRUENCE AMONG LINE SEGMENTS
When are two line segments congruent?  Observe the two pairs of line segments given
here (Fig 7.4).
       (i) (ii)
Fig 7.4
Use the ‘trace-copy’ superposition method for the pair of line segments in [Fig 7.4(i)].
Copy CD and place it on AB . Y ou find that CD covers AB , with C on A and D on B.
Hence, the line segments are congruent.   We write AB CD ? .
Repeat this activity for the pair of line segments in [Fig 7.4(ii)]. What do you find?
They are not congruent.  How do you know it?  It is because the line segments do not
coincide when placed one over other.
Y ou should have by now noticed that the pair of line segments in [Fig 7.4(i)] matched
with each other because they had same length; and this was not the case in [Fig 7.4(ii)].
If two line segments have the same (i.e., equal) length, they are congruent.  Also,
if two line segments are congruent, they have the same length.
2022-23
Page 3


CONGRUENCE OF TRIANGLES 133 133 133 133 133
7.1  INTRODUCTION
Y ou are now ready to learn a very important geometrical idea, Congruence. In particular,
you will study a lot about congruence of triangles.
T o understand what congruence is, we turn to some activities.
T ake two stamps (Fig 7.1) of same denomination. Place one stamp over
the other. What do you observe?
Fig 7.1
One stamp covers the other completely and exactly. This means that the two stamps are
of the same shape and same size. Such objects are said to be congruent. The two stamps
used by you are congruent to one another. Congruent objects are exact copies of one
another.
Can you, now, say if the following objects are congruent or not?
 1. Shaving blades of the same company [Fig 7.2 (i)].
 2. Sheets of the same letter-pad [Fig 7.2 (ii)]. 3. Biscuits in the same packet [Fig 7.2 (iii)].
4. T oys made of the same mould. [Fig 7.2(iv)]
(i) (ii) (iii) (iv)
Chapter  7
Congruence of
Triangles
DO THIS
Fig 7.2
2022-23
MATHEMATICS 134 134 134 134 134
The relation of two objects being congruent is called congruence. For the present,
we will deal with plane figures only, although congruence is a general idea applicable to
three-dimensional shapes also. W e will try to learn a precise meaning of the congruence
of plane figures already known.
7.2  CONGRUENCE OF PLANE FIGURES
Look at the two figures given here (Fig 7.3). Are they congruent?
(i) (ii)
Fig 7.3
Y ou can use the method of superposition. T ake a trace-copy of one of them and place
it over the other . If the figures cover each other completely , they are congruent. Alternatively ,
you may cut out one of them and place it over the other. Beware! Y ou are not allowed to
bend, twist or stretch the figure that is cut out (or traced out).
In Fig 7.3, if figure F
1
 is congruent to figure F
2 
, we write F
1
 ? F
2
.
7.3  CONGRUENCE AMONG LINE SEGMENTS
When are two line segments congruent?  Observe the two pairs of line segments given
here (Fig 7.4).
       (i) (ii)
Fig 7.4
Use the ‘trace-copy’ superposition method for the pair of line segments in [Fig 7.4(i)].
Copy CD and place it on AB . Y ou find that CD covers AB , with C on A and D on B.
Hence, the line segments are congruent.   We write AB CD ? .
Repeat this activity for the pair of line segments in [Fig 7.4(ii)]. What do you find?
They are not congruent.  How do you know it?  It is because the line segments do not
coincide when placed one over other.
Y ou should have by now noticed that the pair of line segments in [Fig 7.4(i)] matched
with each other because they had same length; and this was not the case in [Fig 7.4(ii)].
If two line segments have the same (i.e., equal) length, they are congruent.  Also,
if two line segments are congruent, they have the same length.
2022-23
CONGRUENCE OF TRIANGLES 135 135 135 135 135
In view of the above fact, when two line segments are congruent, we sometimes just
say that the line segments are equal; and we also write AB = CD. (What we actually mean
is AB ? CD ).
7.4  CONGRUENCE OF ANGLES
Look at the four angles given here (Fig 7.5).
(i) (ii)            (iii)        (iv)
Fig 7.5
Make a trace-copy of ?PQR. Try to superpose it on ?ABC. For this, first place
Q on B and QP
    
 along BA
u r uu
. Where does QR
u r uu
fall? It falls onBC
u r uu
.
Thus, ?PQR matches exactly with ?ABC.
That is, ?ABC and ?PQR are congruent.
(Note that the measurement of these two congruent angles are same).
W e write ?ABC ? ?PQR (i)
or m?ABC = m ?PQR(In this case, measure is 40°).
Now, you take a trace-copy of ?LMN. Try to superpose it on ?ABC. Place M on B
and ML
u r uu
 along BA
u r uu
. Does MN
u r uuu
 fall on BC
u r uu
? No, in this case it does not happen. Y ou find
that ?ABC and ?LMN do not cover each other exactly. So, they are not congruent.
(Note that, in this case, the measures of ?ABC and ?LMN are not equal).
What about angles ?XYZ and ?ABC? The raysYX
u r uu
 and YZ
u r uu
, respectively appear
[in Fig 7.5 (iv)] to be longer than BA
u r uu
 and 
BC
u r uu
. Y ou may, hence, think that ?ABC is
‘smaller’ than ?XYZ. But remember that the rays in the figure only indicate the direction
and not any length. On superposition, you will find that these two angles are also congruent.
W e write ?ABC ? ?XYZ (ii)
or m?ABC = m?XYZ
In view of (i) and (ii), we may even write
?ABC ? ?PQR ? ?XYZ
If two angles have the same measure, they are congruent. Also, if two angles are
congruent, their measures are same.
2022-23
Page 4


CONGRUENCE OF TRIANGLES 133 133 133 133 133
7.1  INTRODUCTION
Y ou are now ready to learn a very important geometrical idea, Congruence. In particular,
you will study a lot about congruence of triangles.
T o understand what congruence is, we turn to some activities.
T ake two stamps (Fig 7.1) of same denomination. Place one stamp over
the other. What do you observe?
Fig 7.1
One stamp covers the other completely and exactly. This means that the two stamps are
of the same shape and same size. Such objects are said to be congruent. The two stamps
used by you are congruent to one another. Congruent objects are exact copies of one
another.
Can you, now, say if the following objects are congruent or not?
 1. Shaving blades of the same company [Fig 7.2 (i)].
 2. Sheets of the same letter-pad [Fig 7.2 (ii)]. 3. Biscuits in the same packet [Fig 7.2 (iii)].
4. T oys made of the same mould. [Fig 7.2(iv)]
(i) (ii) (iii) (iv)
Chapter  7
Congruence of
Triangles
DO THIS
Fig 7.2
2022-23
MATHEMATICS 134 134 134 134 134
The relation of two objects being congruent is called congruence. For the present,
we will deal with plane figures only, although congruence is a general idea applicable to
three-dimensional shapes also. W e will try to learn a precise meaning of the congruence
of plane figures already known.
7.2  CONGRUENCE OF PLANE FIGURES
Look at the two figures given here (Fig 7.3). Are they congruent?
(i) (ii)
Fig 7.3
Y ou can use the method of superposition. T ake a trace-copy of one of them and place
it over the other . If the figures cover each other completely , they are congruent. Alternatively ,
you may cut out one of them and place it over the other. Beware! Y ou are not allowed to
bend, twist or stretch the figure that is cut out (or traced out).
In Fig 7.3, if figure F
1
 is congruent to figure F
2 
, we write F
1
 ? F
2
.
7.3  CONGRUENCE AMONG LINE SEGMENTS
When are two line segments congruent?  Observe the two pairs of line segments given
here (Fig 7.4).
       (i) (ii)
Fig 7.4
Use the ‘trace-copy’ superposition method for the pair of line segments in [Fig 7.4(i)].
Copy CD and place it on AB . Y ou find that CD covers AB , with C on A and D on B.
Hence, the line segments are congruent.   We write AB CD ? .
Repeat this activity for the pair of line segments in [Fig 7.4(ii)]. What do you find?
They are not congruent.  How do you know it?  It is because the line segments do not
coincide when placed one over other.
Y ou should have by now noticed that the pair of line segments in [Fig 7.4(i)] matched
with each other because they had same length; and this was not the case in [Fig 7.4(ii)].
If two line segments have the same (i.e., equal) length, they are congruent.  Also,
if two line segments are congruent, they have the same length.
2022-23
CONGRUENCE OF TRIANGLES 135 135 135 135 135
In view of the above fact, when two line segments are congruent, we sometimes just
say that the line segments are equal; and we also write AB = CD. (What we actually mean
is AB ? CD ).
7.4  CONGRUENCE OF ANGLES
Look at the four angles given here (Fig 7.5).
(i) (ii)            (iii)        (iv)
Fig 7.5
Make a trace-copy of ?PQR. Try to superpose it on ?ABC. For this, first place
Q on B and QP
    
 along BA
u r uu
. Where does QR
u r uu
fall? It falls onBC
u r uu
.
Thus, ?PQR matches exactly with ?ABC.
That is, ?ABC and ?PQR are congruent.
(Note that the measurement of these two congruent angles are same).
W e write ?ABC ? ?PQR (i)
or m?ABC = m ?PQR(In this case, measure is 40°).
Now, you take a trace-copy of ?LMN. Try to superpose it on ?ABC. Place M on B
and ML
u r uu
 along BA
u r uu
. Does MN
u r uuu
 fall on BC
u r uu
? No, in this case it does not happen. Y ou find
that ?ABC and ?LMN do not cover each other exactly. So, they are not congruent.
(Note that, in this case, the measures of ?ABC and ?LMN are not equal).
What about angles ?XYZ and ?ABC? The raysYX
u r uu
 and YZ
u r uu
, respectively appear
[in Fig 7.5 (iv)] to be longer than BA
u r uu
 and 
BC
u r uu
. Y ou may, hence, think that ?ABC is
‘smaller’ than ?XYZ. But remember that the rays in the figure only indicate the direction
and not any length. On superposition, you will find that these two angles are also congruent.
W e write ?ABC ? ?XYZ (ii)
or m?ABC = m?XYZ
In view of (i) and (ii), we may even write
?ABC ? ?PQR ? ?XYZ
If two angles have the same measure, they are congruent. Also, if two angles are
congruent, their measures are same.
2022-23
MATHEMATICS 136 136 136 136 136
As in the case of line segments, congruency of angles entirely depends on the equality
of their measures. So, to say that two angles are congruent, we sometimes just say that the
angles are equal; and we write
?ABC =  ?PQR  (to mean ?ABC ? ?PQR).
7.5  CONGRUENCE OF TRIANGLES
We saw that two line segments are congruent where one of them, is just a copy of the
other. Similarly, two angles are congruent if one of them is a copy of the other. W e extend
this idea to triangles.
Two triangles are congruent if they are copies of each other and when superposed,
they cover each other exactly.
(i) (ii)
Fig 7.6
?ABC and ?PQR have the same size and shape. They are congruent. So, we would
express this as
?ABC ? ?PQR
This means that, when you place ?PQR on ?ABC, P falls on  A, Q falls on B and R
falls on C, also 
PQ
 falls along AB , QR falls along BC and PR falls along AC . If, under
a given correspondence, two triangles are congruent, then their corresponding parts
(i.e., angles and sides) that match one another are equal. Thus, in these two congruent
triangles, we have:
Corresponding vertices : A and P , B and Q, C and R.
Corresponding sides :
AB
and PQ , BC and QR , AC and PR .
Corresponding angles : ?A and ?P ,   ?B and ?Q, ?C and ?R.
If you place ?PQR on ?ABC such that P falls on B, then, should the other vertices
also correspond suitably? It need not happen! T ake trace, copies of the triangles and try
to find out.
This shows that while talking about congruence of triangles, not only the measures of
angles and lengths of sides matter, but also the matching of vertices. In the above case, the
correspondence is
A ? P , B ? Q, C ? R
We may write this as ABC ? PQR
A
B C
2022-23
Page 5


CONGRUENCE OF TRIANGLES 133 133 133 133 133
7.1  INTRODUCTION
Y ou are now ready to learn a very important geometrical idea, Congruence. In particular,
you will study a lot about congruence of triangles.
T o understand what congruence is, we turn to some activities.
T ake two stamps (Fig 7.1) of same denomination. Place one stamp over
the other. What do you observe?
Fig 7.1
One stamp covers the other completely and exactly. This means that the two stamps are
of the same shape and same size. Such objects are said to be congruent. The two stamps
used by you are congruent to one another. Congruent objects are exact copies of one
another.
Can you, now, say if the following objects are congruent or not?
 1. Shaving blades of the same company [Fig 7.2 (i)].
 2. Sheets of the same letter-pad [Fig 7.2 (ii)]. 3. Biscuits in the same packet [Fig 7.2 (iii)].
4. T oys made of the same mould. [Fig 7.2(iv)]
(i) (ii) (iii) (iv)
Chapter  7
Congruence of
Triangles
DO THIS
Fig 7.2
2022-23
MATHEMATICS 134 134 134 134 134
The relation of two objects being congruent is called congruence. For the present,
we will deal with plane figures only, although congruence is a general idea applicable to
three-dimensional shapes also. W e will try to learn a precise meaning of the congruence
of plane figures already known.
7.2  CONGRUENCE OF PLANE FIGURES
Look at the two figures given here (Fig 7.3). Are they congruent?
(i) (ii)
Fig 7.3
Y ou can use the method of superposition. T ake a trace-copy of one of them and place
it over the other . If the figures cover each other completely , they are congruent. Alternatively ,
you may cut out one of them and place it over the other. Beware! Y ou are not allowed to
bend, twist or stretch the figure that is cut out (or traced out).
In Fig 7.3, if figure F
1
 is congruent to figure F
2 
, we write F
1
 ? F
2
.
7.3  CONGRUENCE AMONG LINE SEGMENTS
When are two line segments congruent?  Observe the two pairs of line segments given
here (Fig 7.4).
       (i) (ii)
Fig 7.4
Use the ‘trace-copy’ superposition method for the pair of line segments in [Fig 7.4(i)].
Copy CD and place it on AB . Y ou find that CD covers AB , with C on A and D on B.
Hence, the line segments are congruent.   We write AB CD ? .
Repeat this activity for the pair of line segments in [Fig 7.4(ii)]. What do you find?
They are not congruent.  How do you know it?  It is because the line segments do not
coincide when placed one over other.
Y ou should have by now noticed that the pair of line segments in [Fig 7.4(i)] matched
with each other because they had same length; and this was not the case in [Fig 7.4(ii)].
If two line segments have the same (i.e., equal) length, they are congruent.  Also,
if two line segments are congruent, they have the same length.
2022-23
CONGRUENCE OF TRIANGLES 135 135 135 135 135
In view of the above fact, when two line segments are congruent, we sometimes just
say that the line segments are equal; and we also write AB = CD. (What we actually mean
is AB ? CD ).
7.4  CONGRUENCE OF ANGLES
Look at the four angles given here (Fig 7.5).
(i) (ii)            (iii)        (iv)
Fig 7.5
Make a trace-copy of ?PQR. Try to superpose it on ?ABC. For this, first place
Q on B and QP
    
 along BA
u r uu
. Where does QR
u r uu
fall? It falls onBC
u r uu
.
Thus, ?PQR matches exactly with ?ABC.
That is, ?ABC and ?PQR are congruent.
(Note that the measurement of these two congruent angles are same).
W e write ?ABC ? ?PQR (i)
or m?ABC = m ?PQR(In this case, measure is 40°).
Now, you take a trace-copy of ?LMN. Try to superpose it on ?ABC. Place M on B
and ML
u r uu
 along BA
u r uu
. Does MN
u r uuu
 fall on BC
u r uu
? No, in this case it does not happen. Y ou find
that ?ABC and ?LMN do not cover each other exactly. So, they are not congruent.
(Note that, in this case, the measures of ?ABC and ?LMN are not equal).
What about angles ?XYZ and ?ABC? The raysYX
u r uu
 and YZ
u r uu
, respectively appear
[in Fig 7.5 (iv)] to be longer than BA
u r uu
 and 
BC
u r uu
. Y ou may, hence, think that ?ABC is
‘smaller’ than ?XYZ. But remember that the rays in the figure only indicate the direction
and not any length. On superposition, you will find that these two angles are also congruent.
W e write ?ABC ? ?XYZ (ii)
or m?ABC = m?XYZ
In view of (i) and (ii), we may even write
?ABC ? ?PQR ? ?XYZ
If two angles have the same measure, they are congruent. Also, if two angles are
congruent, their measures are same.
2022-23
MATHEMATICS 136 136 136 136 136
As in the case of line segments, congruency of angles entirely depends on the equality
of their measures. So, to say that two angles are congruent, we sometimes just say that the
angles are equal; and we write
?ABC =  ?PQR  (to mean ?ABC ? ?PQR).
7.5  CONGRUENCE OF TRIANGLES
We saw that two line segments are congruent where one of them, is just a copy of the
other. Similarly, two angles are congruent if one of them is a copy of the other. W e extend
this idea to triangles.
Two triangles are congruent if they are copies of each other and when superposed,
they cover each other exactly.
(i) (ii)
Fig 7.6
?ABC and ?PQR have the same size and shape. They are congruent. So, we would
express this as
?ABC ? ?PQR
This means that, when you place ?PQR on ?ABC, P falls on  A, Q falls on B and R
falls on C, also 
PQ
 falls along AB , QR falls along BC and PR falls along AC . If, under
a given correspondence, two triangles are congruent, then their corresponding parts
(i.e., angles and sides) that match one another are equal. Thus, in these two congruent
triangles, we have:
Corresponding vertices : A and P , B and Q, C and R.
Corresponding sides :
AB
and PQ , BC and QR , AC and PR .
Corresponding angles : ?A and ?P ,   ?B and ?Q, ?C and ?R.
If you place ?PQR on ?ABC such that P falls on B, then, should the other vertices
also correspond suitably? It need not happen! T ake trace, copies of the triangles and try
to find out.
This shows that while talking about congruence of triangles, not only the measures of
angles and lengths of sides matter, but also the matching of vertices. In the above case, the
correspondence is
A ? P , B ? Q, C ? R
We may write this as ABC ? PQR
A
B C
2022-23
CONGRUENCE OF TRIANGLES 137 137 137 137 137
EXAMPLE 1 ?ABC and ?PQR are congruent under the correspondence:
ABC ? RQP
Write the parts of ?ABC that correspond to
(i) (ii) ?Q (iii) RP
SOLUTION  For better understanding of the correspondence, let us use a diagram (Fig 7.7).
Fig 7.7
The correspondence is ABC ? RQP . This means
A ? R  ; B ? Q; and C ? P.
So, (i) PQ ? CB (ii) ?Q ? ?B   and (iii) RP ? AC
THINK, DISCUSS AND WRITE
When two triangles, say ABC and PQR are given, there are, in all, six possible
matchings or correspondences. Two of them are
(i) ABC ? PQR and (ii) ABC ? QRP .
Find the other four correspondences by using two cutouts of triangles. Will all
these correspondences lead to congruence? Think about it.
EXERCISE 7.1
1. Complete the following statements:
(a) Two line segments are congruent if ___________.
(b) Among two congruent angles, one has a measure of 70°; the measure of
the other angle is ___________.
(c) When we write  ?A = ?B, we actually mean ___________.
2. Give any two real-life examples for congruent shapes.
3. If ?ABC ? ?FED under the correspondence ABC ? FED, write all the
corresponding congruent parts of the triangles.
4. If ?DEF ? ?BCA, write the part(s) of  ?BCA that correspond to
(i) ?E (ii) EF (iii) ?F (iv) DF
PQ
2022-23