In a parallelogram $ABCD$,$\angle A = 5x - 40^{\circ}$ and $\angle C = 3x + 10^{\circ}$,then $x = \ldots$

In parallelogram $PQRS$,diagonal $PR$ bisects $\angle P$. Prove that diagonal $PR$ bisects $\angle R$ also and $PQRS$ is a rhombus.

In parallelogram $ABCD$,$\angle A = \angle B + 50^{\circ}$. Find the measure of each angle of the quadrilateral $ABCD$.

$P, Q, R$ and $S$ are respectively the mid-points of the sides $AB, BC, CD$ and $DA$ of a quadrilateral $ABCD$ in which $AC = BD$. Prove that $PQRS$ is a rhombus.

$D, E$ and $F$ are the mid-points of the sides $BC, CA$ and $AB$ respectively of an equilateral triangle $ABC$. Show that $\triangle DEF$ is also an equilateral triangle.

$D$ and $E$ are the mid-points of the sides $AB$ and $AC$ respectively of $\triangle ABC$. $DE$ is produced to $F$. To prove that $CF$ is equal and parallel to $DA$,we need an additional information which is