Question

Two sides $AB,BC$ and medium $AM$ of $△ABC$ are congruent to the two sides and the median of $△PQR$. Prove $△ABC\cong △PQR$

Answer 1

Given:

$AM$ is the median of $△ABC$ & $PN$ is the median of $△PQR$.

$AB=PQ,BC=QR$ & $AM=PN$

Since $AM$ & $PN$ is the median of $△ABC$

$\frac{1}{2}BC=BM$ & $\frac{1}{2}QR=QN$ ------ (AM and PN are median)

Now,

$BC=QR$ ----- (given)

$\frac{1}{2}BC=\frac{1}{2}QR$ ------ (Divide both sides by 2)

$⟹BM=QN$

In $△ABM$ and $△PQN$,

$AM=PN$ ------ (Given)

$AB=PQ$ ------ (Given)

$BM=QN$ ------ (Proved above)

$\therefore △ABM\cong △PQN$ ------ (by SSS congruence rule)

$\angle B=\angle Q$ ------ (CPCT)

In $△ABC$ and $△PQR$,

$AB=PQ$ ------ (Given)

$\angle B=\angle Q$ ------ (proved above)

$BC=QR$ ----- (Given)

$\therefore △ABC\cong △PQR$ ------ ( by SAS congruence rule)