$PQRS$ is a parallelogram. If the two diagonals are equal,find the measure of $\angle PQR$. (in $^{\circ}$)

The line segment joining the mid-points $M$ and $N$ of the parallel sides $AB$ and $DC$ respectively of a trapezium $ABCD$ is perpendicular to both the sides $AB$ and $DC$. Prove that $AD = BC$.

In the figure,$PQ > PR$ and $QS$ and $RS$ are the bisectors of $\angle Q$ and $\angle R$,respectively. Show that $SQ > SR$.

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

$M$ is a point on side $BC$ of a triangle $ABC$ such that $AM$ is the bisector of $\angle BAC$. Is it true to say that the perimeter of the triangle is greater than $2\, AM$? Give reason for your answer.

In $\Delta ABC$ and $\Delta PQR$,if $\frac{AB}{PQ} = \frac{BC}{RQ} = \frac{AC}{PR} = 1$,then $\Delta ABC \cong \Delta \ldots \ldots \ldots$