Question

Q.12. Two sides AB and BC and the median AD of A ABC are equal respectively to the two sides PQ
and QR and the median PM of the other A POR. Prove that the two triangles ABC and PQR are
congruent.​

Answer 1

Given : (i) AB = PQ

(ii) BC = QR

(iii) AD = PM

To Prove : triangle ABC is congruent to triangle PQR

Proof :

Since BC = QR and they are divided by median thus, BC/2 = QR/2, that is, BD = QM

In triangle ABD and triangle PQM

AB = PQ

BD = QM

AD = PM

Thus, triangle ABD is congruent to triangle PQM

{ by SSS rule}

angle B = angle Q (by C.P.C.T.).........(i)

In triangle ABC and triangle PQR

AB = PQ

BC = QR

angle B = angle Q {using (i)}

Thus, triangle ABC is congruent to triangle PQR

{by SAS rule}

Hence, PROVED

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Answer 2

Step-by-step explanation:

△ABC and△PQR in which AB=PQ,BC=QR and AM=PN. 

Since AM and PN are median of triangles ABC and PQR respectively. 

Now, BC=QR ∣  Given 

⇒21BC=21QR ∣ Median divides opposite sides in two equal parts

BM=QN... (1) 

Now, in △ABM and△PQN we have 

AB=PQ  ∣ Given

BM=QN ∣  From (i)

and AM=PN ∣  Given

∴ By SSS criterion of congruence, we have 

△ABM≅△PQN, which proves (i) 

∠B=∠Q ... (2)  ∣ Since, corresponding parts of the congruent triangle are equal

Now, in  △ABC and△PQR we have 

AB=PQ ∣ Given

∠B=∠Q ∣ From (2)

BC=QR ∣ Given

∴ by SAS criterion of congruence, we have 

△ABC≅△PQR, which proves (ii)