For the correspondence $ABC \leftrightarrow PRQ$ between $\Delta ABC$ and $\Delta PQR,$ the side $\ldots$ corresponds to $AB.$

$PL, QM$ and $RN$ are the altitudes of $\Delta PQR$. If $PL = QM = RN$,then by $AAS$ criterion of congruence,prove that $\Delta PQR$ is an equilateral triangle.

In triangles $ABC$ and $DEF$,$\angle A = \angle D$,$\angle B = \angle E$ and $AB = EF$. Will the two triangles be congruent? Give reasons for your answer.

In the given figure,$AM$ and $BN$ are both perpendicular to $AB$. $MN$ intersects $AB$ at $P$. Also,$P$ is the midpoint of $AB$. Prove that $AM = BN$ and $P$ is the midpoint of $MN$.

Point $P$ lies in the interior of $\Delta ABC$. Prove that $PB + PC < AB + AC$.

If the bisector of an angle of a triangle also bisects the opposite side,prove that the triangle is isosceles.