In $\Delta ABC$ and $\Delta PQR$,if $\angle B = \angle P = 90^{\circ}$,$AC = RQ$,and $AB = RP$,then $\Delta ABC \cong \Delta \ldots \ldots \ldots$

Two lines $l$ and $m$ intersect at the point $O$ and $P$ is a point on a line $n$ passing through the point $O$ such that $P$ is equidistant from $l$ and $m$. Prove that $n$ is the bisector of the angle formed by $l$ and $m$.

In quadrilateral $PQRS$,$PQ = PS$ and $RQ = RS$. Prove that $\angle PQR = \angle PSR$.

Is it possible to construct a triangle with lengths of its sides as $8\,cm, 7\,cm$ and $4\,cm$? Give reason for your answer.

In $\Delta ABC$,the bisectors of $\angle B$ and $\angle C$ intersect at $P$. $A$ line drawn through $P$ and parallel to $BC$ intersects $AB$ at $X$ and $AC$ at $Y$. Prove that $XY = XB + YC$.

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 $MN$.