Question

In given figures two sides $AB$ and $BC$ and median $AM$ of one triangle $ABC$ are respectively equal to sides $PQ$ and $QR$ and median $PN$ of $△PQR$. Show that:
(i) $△ABM\cong △PQN$
(ii) $△ABC\cong △PQR$

Answer 1

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

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

Now, $BC=QR$ $\mid$ Given

$\Rightarrow \frac{1}{2}BC=\frac{1}{2}QR$ $\mid$ Median divides opposite sides in two equal parts

$BM=QN$... (1)

Now, in $△ABM$ and$△PQN$ we have

$AB=PQ$ $\mid$ Given

$BM=QN$ $\mid$ From (i)

and $AM=PN$ $\mid$ Given

$\therefore$ By SSS criterion of congruence, we have

$△ABM\cong △PQN,$ which proves (i)

$\angle B=\angle Q$ ... (2) $\mid$ Since, corresponding parts of the congruent triangle are equal

Now, in $△ABC$ and$△PQR$ we have

$AB=PQ$ $\mid$ Given

$\angle B=\angle Q$ $\mid$ From (2)

$BC=QR$ $\mid$ Given

$\therefore$ by SAS criterion of congruence, we have

$△ABC\cong △PQR$, which proves (ii)