Question

In the adjoining figure, two sides AB, AC and altitude AM of ΔABC are
respectively equal to two sides PQ, PR and altitude PN of ΔPQR. Prove
that ΔABC ≅ ΔPQR​

Answer 1

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△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 

⇒½BC=½QR ∣ 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)

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Answer 2

Answer:

❥Answer

❥Answer

△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

⇒½BC=½QR ∣ 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)

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