Congruence (Exercise 16.2) RD Sharma Solutions | Mathematics (Maths) Class 7 PDF Download

 Page 1


 
 
 
 
  
 
 
 
        
 
1. In the following pairs of triangle (Fig. 12 to 15), the lengths of the sides are indicated 
along sides. By applying SSS condition, determine which are congruent. State the 
result in symbolic form. 
 
 
 
 
 
Page 2


 
 
 
 
  
 
 
 
        
 
1. In the following pairs of triangle (Fig. 12 to 15), the lengths of the sides are indicated 
along sides. By applying SSS condition, determine which are congruent. State the 
result in symbolic form. 
 
 
 
 
 
 
 
 
 
  
 
 
 
Solution: 
(i) In ? ABC and ? DEF 
AB = DE = 4.5 cm (Side) 
BC = EF = 6 cm (Side) and 
AC = DF = 4 cm (Side) 
SSS criterion is two triangles are congruent, if the three sides of triangle are respectively 
equal to the three sides of the other triangle.  
Therefore, by SSS criterion of congruence, ?ABC ? ?DEF 
 
(ii) In ? ACB and ? ADB 
AC = AD = 5.5cm (Side) 
BC = BD = 5cm (Side) and 
AB = AB = 6cm (Side) 
SSS criterion is two triangles are congruent, if the three sides of triangle are respectively 
equal to the three sides of the other triangle.  
Therefore, by SSS criterion of congruence, ?ACB ? ?ADB 
 
(iii) In ? ABD and ? FEC, 
AB = FE = 5cm (Side) 
AD = FC = 10.5cm (Side) 
BD = CE = 7cm (Side) 
SSS criterion is two triangles are congruent, if the three sides of triangle are respectively 
equal to the three sides of the other triangle.  
Therefore, by SSS criterion of congruence, ?ABD ? ?FEC 
 
(iv) In ? ABO and ? DOC, 
AB = DC = 4cm (Side) 
AO = OC = 2cm (Side) 
BO = OD = 3.5cm (Side) 
SSS criterion is two triangles are congruent, if the three sides of triangle are respectively 
equal to the three sides of the other triangle.  
Therefore, by SSS criterion of congruence, ?ABO ? ?ODC 
 
2. In fig.16, AD = DC and AB = BC 
(i) Is ?ABD ? ?CBD? 
(ii) State the three parts of matching pairs you have used to answer (i). 
Page 3


 
 
 
 
  
 
 
 
        
 
1. In the following pairs of triangle (Fig. 12 to 15), the lengths of the sides are indicated 
along sides. By applying SSS condition, determine which are congruent. State the 
result in symbolic form. 
 
 
 
 
 
 
 
 
 
  
 
 
 
Solution: 
(i) In ? ABC and ? DEF 
AB = DE = 4.5 cm (Side) 
BC = EF = 6 cm (Side) and 
AC = DF = 4 cm (Side) 
SSS criterion is two triangles are congruent, if the three sides of triangle are respectively 
equal to the three sides of the other triangle.  
Therefore, by SSS criterion of congruence, ?ABC ? ?DEF 
 
(ii) In ? ACB and ? ADB 
AC = AD = 5.5cm (Side) 
BC = BD = 5cm (Side) and 
AB = AB = 6cm (Side) 
SSS criterion is two triangles are congruent, if the three sides of triangle are respectively 
equal to the three sides of the other triangle.  
Therefore, by SSS criterion of congruence, ?ACB ? ?ADB 
 
(iii) In ? ABD and ? FEC, 
AB = FE = 5cm (Side) 
AD = FC = 10.5cm (Side) 
BD = CE = 7cm (Side) 
SSS criterion is two triangles are congruent, if the three sides of triangle are respectively 
equal to the three sides of the other triangle.  
Therefore, by SSS criterion of congruence, ?ABD ? ?FEC 
 
(iv) In ? ABO and ? DOC, 
AB = DC = 4cm (Side) 
AO = OC = 2cm (Side) 
BO = OD = 3.5cm (Side) 
SSS criterion is two triangles are congruent, if the three sides of triangle are respectively 
equal to the three sides of the other triangle.  
Therefore, by SSS criterion of congruence, ?ABO ? ?ODC 
 
2. In fig.16, AD = DC and AB = BC 
(i) Is ?ABD ? ?CBD? 
(ii) State the three parts of matching pairs you have used to answer (i). 
 
 
 
 
  
 
 
 
 
Solution: 
(i) Yes ?ABD ??CBD by the SSS criterion.  
SSS criterion is two triangles are congruent, if the three sides of triangle are respectively 
equal to the three sides of the other triangle.  
Hence ?ABD ??CBD 
 
(ii) We have used the three conditions in the SSS criterion as follows: 
AD = DC 
AB = BC and 
DB = BD 
 
3. In Fig. 17, AB = DC and BC = AD. 
(i) Is ?ABC ? ?CDA? 
(ii) What congruence condition have you used? 
(iii) You have used some fact, not given in the question, what is that? 
 
Solution: 
(i) From the figure we have AB = DC 
BC = AD 
Page 4


 
 
 
 
  
 
 
 
        
 
1. In the following pairs of triangle (Fig. 12 to 15), the lengths of the sides are indicated 
along sides. By applying SSS condition, determine which are congruent. State the 
result in symbolic form. 
 
 
 
 
 
 
 
 
 
  
 
 
 
Solution: 
(i) In ? ABC and ? DEF 
AB = DE = 4.5 cm (Side) 
BC = EF = 6 cm (Side) and 
AC = DF = 4 cm (Side) 
SSS criterion is two triangles are congruent, if the three sides of triangle are respectively 
equal to the three sides of the other triangle.  
Therefore, by SSS criterion of congruence, ?ABC ? ?DEF 
 
(ii) In ? ACB and ? ADB 
AC = AD = 5.5cm (Side) 
BC = BD = 5cm (Side) and 
AB = AB = 6cm (Side) 
SSS criterion is two triangles are congruent, if the three sides of triangle are respectively 
equal to the three sides of the other triangle.  
Therefore, by SSS criterion of congruence, ?ACB ? ?ADB 
 
(iii) In ? ABD and ? FEC, 
AB = FE = 5cm (Side) 
AD = FC = 10.5cm (Side) 
BD = CE = 7cm (Side) 
SSS criterion is two triangles are congruent, if the three sides of triangle are respectively 
equal to the three sides of the other triangle.  
Therefore, by SSS criterion of congruence, ?ABD ? ?FEC 
 
(iv) In ? ABO and ? DOC, 
AB = DC = 4cm (Side) 
AO = OC = 2cm (Side) 
BO = OD = 3.5cm (Side) 
SSS criterion is two triangles are congruent, if the three sides of triangle are respectively 
equal to the three sides of the other triangle.  
Therefore, by SSS criterion of congruence, ?ABO ? ?ODC 
 
2. In fig.16, AD = DC and AB = BC 
(i) Is ?ABD ? ?CBD? 
(ii) State the three parts of matching pairs you have used to answer (i). 
 
 
 
 
  
 
 
 
 
Solution: 
(i) Yes ?ABD ??CBD by the SSS criterion.  
SSS criterion is two triangles are congruent, if the three sides of triangle are respectively 
equal to the three sides of the other triangle.  
Hence ?ABD ??CBD 
 
(ii) We have used the three conditions in the SSS criterion as follows: 
AD = DC 
AB = BC and 
DB = BD 
 
3. In Fig. 17, AB = DC and BC = AD. 
(i) Is ?ABC ? ?CDA? 
(ii) What congruence condition have you used? 
(iii) You have used some fact, not given in the question, what is that? 
 
Solution: 
(i) From the figure we have AB = DC 
BC = AD 
 
 
 
 
  
 
 
 
And AC = CA 
SSS criterion is two triangles are congruent, if the three sides of triangle are respectively 
equal to the three sides of the other triangle.  
Therefore by SSS criterion ?ABC ? ?CDA 
 
(ii) We have used Side Side Side congruence condition with one side common in both 
the triangles. 
 
(iii)Yes, we have used the fact that AC = CA. 
 
4. In ?PQR ? ?EFD, 
(i) Which side of ?PQR equals ED? 
(ii) Which angle of ?PQR equals angle E? 
 
Solution: 
 
(i) PR = ED  
Since the corresponding sides of congruent triangles are equal. 
 
(ii) ?QPR = ?FED  
Since the corresponding angles of congruent triangles are equal. 
 
5. Triangles ABC and PQR are both isosceles with AB = AC and PQ = PR respectively. If 
also, AB = PQ and BC = QR, are the two triangles congruent? Which condition do you 
use? 
It ?B = 50°, what is the measure of ?R? 
 
Solution: 
Given that AB = AC in isosceles ?ABC 
And PQ = PR in isosceles ?PQR. 
Also given that AB = PQ and QR = BC. 
Page 5


 
 
 
 
  
 
 
 
        
 
1. In the following pairs of triangle (Fig. 12 to 15), the lengths of the sides are indicated 
along sides. By applying SSS condition, determine which are congruent. State the 
result in symbolic form. 
 
 
 
 
 
 
 
 
 
  
 
 
 
Solution: 
(i) In ? ABC and ? DEF 
AB = DE = 4.5 cm (Side) 
BC = EF = 6 cm (Side) and 
AC = DF = 4 cm (Side) 
SSS criterion is two triangles are congruent, if the three sides of triangle are respectively 
equal to the three sides of the other triangle.  
Therefore, by SSS criterion of congruence, ?ABC ? ?DEF 
 
(ii) In ? ACB and ? ADB 
AC = AD = 5.5cm (Side) 
BC = BD = 5cm (Side) and 
AB = AB = 6cm (Side) 
SSS criterion is two triangles are congruent, if the three sides of triangle are respectively 
equal to the three sides of the other triangle.  
Therefore, by SSS criterion of congruence, ?ACB ? ?ADB 
 
(iii) In ? ABD and ? FEC, 
AB = FE = 5cm (Side) 
AD = FC = 10.5cm (Side) 
BD = CE = 7cm (Side) 
SSS criterion is two triangles are congruent, if the three sides of triangle are respectively 
equal to the three sides of the other triangle.  
Therefore, by SSS criterion of congruence, ?ABD ? ?FEC 
 
(iv) In ? ABO and ? DOC, 
AB = DC = 4cm (Side) 
AO = OC = 2cm (Side) 
BO = OD = 3.5cm (Side) 
SSS criterion is two triangles are congruent, if the three sides of triangle are respectively 
equal to the three sides of the other triangle.  
Therefore, by SSS criterion of congruence, ?ABO ? ?ODC 
 
2. In fig.16, AD = DC and AB = BC 
(i) Is ?ABD ? ?CBD? 
(ii) State the three parts of matching pairs you have used to answer (i). 
 
 
 
 
  
 
 
 
 
Solution: 
(i) Yes ?ABD ??CBD by the SSS criterion.  
SSS criterion is two triangles are congruent, if the three sides of triangle are respectively 
equal to the three sides of the other triangle.  
Hence ?ABD ??CBD 
 
(ii) We have used the three conditions in the SSS criterion as follows: 
AD = DC 
AB = BC and 
DB = BD 
 
3. In Fig. 17, AB = DC and BC = AD. 
(i) Is ?ABC ? ?CDA? 
(ii) What congruence condition have you used? 
(iii) You have used some fact, not given in the question, what is that? 
 
Solution: 
(i) From the figure we have AB = DC 
BC = AD 
 
 
 
 
  
 
 
 
And AC = CA 
SSS criterion is two triangles are congruent, if the three sides of triangle are respectively 
equal to the three sides of the other triangle.  
Therefore by SSS criterion ?ABC ? ?CDA 
 
(ii) We have used Side Side Side congruence condition with one side common in both 
the triangles. 
 
(iii)Yes, we have used the fact that AC = CA. 
 
4. In ?PQR ? ?EFD, 
(i) Which side of ?PQR equals ED? 
(ii) Which angle of ?PQR equals angle E? 
 
Solution: 
 
(i) PR = ED  
Since the corresponding sides of congruent triangles are equal. 
 
(ii) ?QPR = ?FED  
Since the corresponding angles of congruent triangles are equal. 
 
5. Triangles ABC and PQR are both isosceles with AB = AC and PQ = PR respectively. If 
also, AB = PQ and BC = QR, are the two triangles congruent? Which condition do you 
use? 
It ?B = 50°, what is the measure of ?R? 
 
Solution: 
Given that AB = AC in isosceles ?ABC 
And PQ = PR in isosceles ?PQR. 
Also given that AB = PQ and QR = BC. 
  
Therefore, AC = PR (AB = AC, PQ = PR and AB = PQ) 
Hence, ?ABC ? ?PQR 
Now 
?ABC = ?PQR (Since triangles are congruent) 
However, ?PQR is isosceles. 
Therefore, ?PRQ = ?PQR = ?ABC = 50
o
6. ABC and DBC are both isosceles triangles on a common base BC such that A and D 
lie on the same side of BC. Are triangles ADB and ADC congruent? Which condition do 
you use? If ?BAC = 40° and ?BDC = 100°, then find ?ADB.
Solution:
Given ABC and DBC are both isosceles triangles on a common base BC
?BAD = ?CAD (corresponding parts of congruent triangles)
?BAD + ?CAD = 40
o
/ 2
?BAD = 40
o
/2 =20
o
?ABC + ?BCA + ?BAC = 180
o
 (Angle sum property)
Since ?ABC is an isosceles triangle,
?ABC = ?BCA
?ABC +?ABC + 40
o 
= 180
o
2 ?ABC = 180
o
– 40
o
 = 140
o
?ABC = 140
o
/2 = 70
o
?DBC + ? BCD + ? BDC = 180
o
 (Angle sum property)
Since ?DBC is an isosceles triangle, ? DBC = ?BCD
?DBC + ?DBC + 100
o 
= 180
o
2 ?DBC = 180°– 100
o
 = 80
o
?DBC = 80
o
/2 = 40
o
In ? BAD,
?ABD + ?BAD + ?ADB = 180
o 
(Angle sum property)
30
o
 + 20
o
 + ?ADB = 180
o
 (?ABD = ?ABC – ?DBC),
?ADB = 180
o
- 20
o
– 30
o
?ADB = 130
o 
7. ? ABC and ?ABD are on a common base AB, and AC = BD and BC = AD as shown in 
Fig. 18. Which of the following statements is true?
(i) ?ABC ? ?ABD
(ii) ?ABC ? ?ADB
(iii) ?ABC ? ?BAD