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

In quadrilateral $ABCD$,the angles are in the ratio $\angle A : \angle B : \angle C : \angle D = 2 : 4 : 5 : 7$. Find the measure of each angle of the quadrilateral and state the type of quadrilateral $ABCD$.

State whether each of the following statements is true or false:
$(1)$ $ABCD$ is a parallelogram. If $AB = 12 \text{ cm}$ and $BC = 5 \text{ cm}$,then $AC = 13 \text{ cm}$.
$(2)$ $ABCD$ is a trapezium. If $AB \parallel CD$ and $AB = 10 \text{ cm}$,then $CD = 10 \text{ cm}$.

The perimeter of rectangle $ABCD$ is $112 \, cm$ and $AB : BC = 5 : 3$. Find the length of $AB$ in $cm$.

The quadrilateral formed by joining the mid-points of the sides of a quadrilateral $PQRS,$ taken in order,is a rhombus,if

In $\Delta ABC$,the points $M$ and $N$ are on sides $AB$ and $AC$ respectively such that $AM = \frac{1}{4} AB$ and $AN = \frac{1}{4} AC$. Prove that $MN = \frac{1}{4} BC$.