Congruence (Exercise 16.3) RD Sharma Solutions | Mathematics (Maths) Class 7 PDF Download

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1. By applying SAS congruence condition, state which of the following pairs (Fig. 28) of 
triangle are congruent. State the result in symbolic form 
 
Solution: 
(i) From the figure we have OA = OC and OB = OD and 
?AOB = ?COD which are vertically opposite angles.  
Therefore by SAS condition, ?AOB ??COD 
 
(ii) From the figure we have BD = DC 
?ADB = ?ADC = 90
o
 and AD = DA 
Therefore, by SAS condition, ?ADB ??ADC. 
 
(iii) From the figure we have AB = DC 
?ABD = ?CDB and BD = DB 
Therefore, by SAS condition, ?ABD ??CBD 
 
(iv) We have BC = QR 
ABC = PQR = 90
o 
And AB = PQ 
Page 2


 
 
 
 
  
 
 
 
         
 
1. By applying SAS congruence condition, state which of the following pairs (Fig. 28) of 
triangle are congruent. State the result in symbolic form 
 
Solution: 
(i) From the figure we have OA = OC and OB = OD and 
?AOB = ?COD which are vertically opposite angles.  
Therefore by SAS condition, ?AOB ??COD 
 
(ii) From the figure we have BD = DC 
?ADB = ?ADC = 90
o
 and AD = DA 
Therefore, by SAS condition, ?ADB ??ADC. 
 
(iii) From the figure we have AB = DC 
?ABD = ?CDB and BD = DB 
Therefore, by SAS condition, ?ABD ??CBD 
 
(iv) We have BC = QR 
ABC = PQR = 90
o 
And AB = PQ 
 
 
 
 
  
 
 
 
Therefore, by SAS condition, ?ABC? ?PQR. 
 
2. State the condition by which the following pairs of triangles are congruent. 
 
Solution:   
(i) AB = AD 
BC = CD and AC = CA 
Therefore by SSS condition, ?ABC? ?ADC 
 
(ii) AC = BD 
AD = BC and AB = BA 
Therefore, by SSS condition, ?ABD ? ?BAC 
 
(iii) AB = AD 
?BAC = ?DAC and AC = CA 
Therefore by SAS condition, ?BAC ? ?DAC 
Page 3


 
 
 
 
  
 
 
 
         
 
1. By applying SAS congruence condition, state which of the following pairs (Fig. 28) of 
triangle are congruent. State the result in symbolic form 
 
Solution: 
(i) From the figure we have OA = OC and OB = OD and 
?AOB = ?COD which are vertically opposite angles.  
Therefore by SAS condition, ?AOB ??COD 
 
(ii) From the figure we have BD = DC 
?ADB = ?ADC = 90
o
 and AD = DA 
Therefore, by SAS condition, ?ADB ??ADC. 
 
(iii) From the figure we have AB = DC 
?ABD = ?CDB and BD = DB 
Therefore, by SAS condition, ?ABD ??CBD 
 
(iv) We have BC = QR 
ABC = PQR = 90
o 
And AB = PQ 
 
 
 
 
  
 
 
 
Therefore, by SAS condition, ?ABC? ?PQR. 
 
2. State the condition by which the following pairs of triangles are congruent. 
 
Solution:   
(i) AB = AD 
BC = CD and AC = CA 
Therefore by SSS condition, ?ABC? ?ADC 
 
(ii) AC = BD 
AD = BC and AB = BA 
Therefore, by SSS condition, ?ABD ? ?BAC 
 
(iii) AB = AD 
?BAC = ?DAC and AC = CA 
Therefore by SAS condition, ?BAC ? ?DAC 
 
 
 
 
  
 
 
 
(iv) AD = BC 
?DAC = ?BCA and AC = CA 
Therefore, by SAS condition, ?ABC ? ?ADC 
 
3. In fig. 30, line segments AB and CD bisect each other at O. Which of the following 
statements is true? 
(i) ?AOC ? ?DOB 
(ii) ?AOC ? ?BOD 
(iii) ?AOC ? ?ODB 
State the three pairs of matching parts, you have used to arrive at the answer. 
 
 
Solution: 
From the figure we have, 
AO = OB 
And, CO = OD 
Also, AOC = BOD 
Therefore, by SAS condition, ?AOC ? ?BOD 
Hence, (ii) statement is true. 
 
4. Line-segments AB and CD bisect each other at O. AC and BD are joined forming 
triangles AOC and BOD. State the three equality relations between the parts of the 
two triangles that are given or otherwise known. Are the two triangles congruent? 
State in symbolic form, which congruence condition do you use? 
 
Solution: 
We have AO = OB and CO = OD  
Since AB and CD bisect each other at 0. 
Also ?AOC = ?BOD  
Since they are opposite angles on the same vertex. 
Page 4


 
 
 
 
  
 
 
 
         
 
1. By applying SAS congruence condition, state which of the following pairs (Fig. 28) of 
triangle are congruent. State the result in symbolic form 
 
Solution: 
(i) From the figure we have OA = OC and OB = OD and 
?AOB = ?COD which are vertically opposite angles.  
Therefore by SAS condition, ?AOB ??COD 
 
(ii) From the figure we have BD = DC 
?ADB = ?ADC = 90
o
 and AD = DA 
Therefore, by SAS condition, ?ADB ??ADC. 
 
(iii) From the figure we have AB = DC 
?ABD = ?CDB and BD = DB 
Therefore, by SAS condition, ?ABD ??CBD 
 
(iv) We have BC = QR 
ABC = PQR = 90
o 
And AB = PQ 
 
 
 
 
  
 
 
 
Therefore, by SAS condition, ?ABC? ?PQR. 
 
2. State the condition by which the following pairs of triangles are congruent. 
 
Solution:   
(i) AB = AD 
BC = CD and AC = CA 
Therefore by SSS condition, ?ABC? ?ADC 
 
(ii) AC = BD 
AD = BC and AB = BA 
Therefore, by SSS condition, ?ABD ? ?BAC 
 
(iii) AB = AD 
?BAC = ?DAC and AC = CA 
Therefore by SAS condition, ?BAC ? ?DAC 
 
 
 
 
  
 
 
 
(iv) AD = BC 
?DAC = ?BCA and AC = CA 
Therefore, by SAS condition, ?ABC ? ?ADC 
 
3. In fig. 30, line segments AB and CD bisect each other at O. Which of the following 
statements is true? 
(i) ?AOC ? ?DOB 
(ii) ?AOC ? ?BOD 
(iii) ?AOC ? ?ODB 
State the three pairs of matching parts, you have used to arrive at the answer. 
 
 
Solution: 
From the figure we have, 
AO = OB 
And, CO = OD 
Also, AOC = BOD 
Therefore, by SAS condition, ?AOC ? ?BOD 
Hence, (ii) statement is true. 
 
4. Line-segments AB and CD bisect each other at O. AC and BD are joined forming 
triangles AOC and BOD. State the three equality relations between the parts of the 
two triangles that are given or otherwise known. Are the two triangles congruent? 
State in symbolic form, which congruence condition do you use? 
 
Solution: 
We have AO = OB and CO = OD  
Since AB and CD bisect each other at 0. 
Also ?AOC = ?BOD  
Since they are opposite angles on the same vertex. 
 
 
 
 
  
 
 
 
Therefore by SAS congruence condition, ?AOC ? ?BOD 
 
5. ?ABC is isosceles with AB = AC. Line segment AD bisects ?A and meets the base BC 
in D. 
 
(i) Is ?ADB ? ?ADC? 
(ii) State the three pairs of matching parts used to answer (i). 
(iii) Is it true to say that BD = DC? 
 
Solution: 
(i) We have AB = AC (Given) 
?BAD = ?CAD (AD bisects ?BAC) 
Therefore by SAS condition of congruence, ?ADB ? ?ADC 
 
(ii) We have used AB, AC; ?BAD = ?CAD; AD, DA. 
 
(iii) Now, ?ADB??ADC  
Therefore by corresponding parts of congruent triangles 
BD = DC. 
 
6. In Fig. 31, AB = AD and ?BAC = ?DAC. 
(i) State in symbolic form the congruence of two triangles ABC and ADC that is true. 
(ii) Complete each of the following, so as to make it true: 
(a) ?ABC = 
(b) ?ACD = 
(c) Line segment AC bisects ….. And …….. 
 
Solution: 
Page 5


 
 
 
 
  
 
 
 
         
 
1. By applying SAS congruence condition, state which of the following pairs (Fig. 28) of 
triangle are congruent. State the result in symbolic form 
 
Solution: 
(i) From the figure we have OA = OC and OB = OD and 
?AOB = ?COD which are vertically opposite angles.  
Therefore by SAS condition, ?AOB ??COD 
 
(ii) From the figure we have BD = DC 
?ADB = ?ADC = 90
o
 and AD = DA 
Therefore, by SAS condition, ?ADB ??ADC. 
 
(iii) From the figure we have AB = DC 
?ABD = ?CDB and BD = DB 
Therefore, by SAS condition, ?ABD ??CBD 
 
(iv) We have BC = QR 
ABC = PQR = 90
o 
And AB = PQ 
 
 
 
 
  
 
 
 
Therefore, by SAS condition, ?ABC? ?PQR. 
 
2. State the condition by which the following pairs of triangles are congruent. 
 
Solution:   
(i) AB = AD 
BC = CD and AC = CA 
Therefore by SSS condition, ?ABC? ?ADC 
 
(ii) AC = BD 
AD = BC and AB = BA 
Therefore, by SSS condition, ?ABD ? ?BAC 
 
(iii) AB = AD 
?BAC = ?DAC and AC = CA 
Therefore by SAS condition, ?BAC ? ?DAC 
 
 
 
 
  
 
 
 
(iv) AD = BC 
?DAC = ?BCA and AC = CA 
Therefore, by SAS condition, ?ABC ? ?ADC 
 
3. In fig. 30, line segments AB and CD bisect each other at O. Which of the following 
statements is true? 
(i) ?AOC ? ?DOB 
(ii) ?AOC ? ?BOD 
(iii) ?AOC ? ?ODB 
State the three pairs of matching parts, you have used to arrive at the answer. 
 
 
Solution: 
From the figure we have, 
AO = OB 
And, CO = OD 
Also, AOC = BOD 
Therefore, by SAS condition, ?AOC ? ?BOD 
Hence, (ii) statement is true. 
 
4. Line-segments AB and CD bisect each other at O. AC and BD are joined forming 
triangles AOC and BOD. State the three equality relations between the parts of the 
two triangles that are given or otherwise known. Are the two triangles congruent? 
State in symbolic form, which congruence condition do you use? 
 
Solution: 
We have AO = OB and CO = OD  
Since AB and CD bisect each other at 0. 
Also ?AOC = ?BOD  
Since they are opposite angles on the same vertex. 
 
 
 
 
  
 
 
 
Therefore by SAS congruence condition, ?AOC ? ?BOD 
 
5. ?ABC is isosceles with AB = AC. Line segment AD bisects ?A and meets the base BC 
in D. 
 
(i) Is ?ADB ? ?ADC? 
(ii) State the three pairs of matching parts used to answer (i). 
(iii) Is it true to say that BD = DC? 
 
Solution: 
(i) We have AB = AC (Given) 
?BAD = ?CAD (AD bisects ?BAC) 
Therefore by SAS condition of congruence, ?ADB ? ?ADC 
 
(ii) We have used AB, AC; ?BAD = ?CAD; AD, DA. 
 
(iii) Now, ?ADB??ADC  
Therefore by corresponding parts of congruent triangles 
BD = DC. 
 
6. In Fig. 31, AB = AD and ?BAC = ?DAC. 
(i) State in symbolic form the congruence of two triangles ABC and ADC that is true. 
(ii) Complete each of the following, so as to make it true: 
(a) ?ABC = 
(b) ?ACD = 
(c) Line segment AC bisects ….. And …….. 
 
Solution: 
 
 
 
 
  
 
 
 
i) AB = AD (given) 
?BAC = ?DAC (given) 
AC = CA (common) 
Therefore by SAS condition of congruency, ?ABC ? ?ADC 
ii) ?ABC = ?ADC (corresponding parts of congruent triangles) 
?ACD = ?ACB (corresponding parts of congruent triangles) 
Line segment AC bisects ?A and ?C. 
 
7. In fig. 32, AB || DC and AB = DC. 
(i) Is ?ACD ? ?CAB? 
(ii) State the three pairs of matching parts used to answer (i). 
(iii) Which angle is equal to ?CAD? 
(iv) Does it follow from (iii) that AD || BC? 
 
Solution: 
(i) Yes by SAS condition of congruency, ?ACD ? ?CAB. 
 
(ii) We have used AB = DC, AC = CA and ?DCA = ?BAC. 
 
(iii) ?CAD = ?ACB since the two triangles are congruent. 
 
(iv) Yes this follows from AD parallel to BC as alternate angles are equal. lf alternate 
angles are equal then the lines are parallel