In the given figure, $P$ is a point in the interior of $\angle A B C$. If $PL \perp BA$ and $PM \perp BC$ such that $PL = PM$, prove that BP is the bisector of $\angle ABC$.
Which of the following pairs of triangles are congruent ? (a) $\triangle ABC$ and $\triangle DEF$ in which : $BC = EF , AC = DF$ and $\angle C =\angle F$. (b) $\triangle ABC$ and $\triangle PQR$ in which : $AB = PQ , BC = QR$ and $\angle C =\angle R$. (c) $\triangle ABC$ and $\triangle LMN$ in which : $\angle A =\angle L =90^{\circ}, AB = LM , \angle C =40^{\circ}$ and $\angle M =50^{\circ}$ (d) $\triangle ABC$ and $\triangle DEF$ in which : $\angle B =\angle E =90^{\circ}$ and $AC = DF$.