Diagonals of a parallelogram $ABCD$ intersect at $O.$ If $\angle BOC = 90^{\circ}$ and $\angle BDC = 50^{\circ},$ then $\angle OAB$ is (in $^{\circ}$)

$ABCD$ is a trapezium such that $AB || CD$. If $\angle A = y + 60^{\circ}$,$\angle B = x + 60^{\circ}$,$\angle C = 3x - 40^{\circ}$ and $\angle D = 3y - 80^{\circ}$,then find the measure of each angle of $ABCD$.

$P$ and $Q$ are the mid-points of the opposite sides $AB$ and $CD$ of a parallelogram $ABCD.$ $AQ$ intersects $DP$ at $S$ and $BQ$ intersects $CP$ at $R.$ Show that $PRQS$ is a parallelogram.

In $\Delta ABC$,$P$ and $Q$ are the midpoints of $AB$ and $AC$ respectively. If $PQ + BC = 8.4 \, cm$ and $AB + AC = 20.5 \, cm$,then find the perimeter of $\Delta ABC$ in $cm$.

$ABCD$ is a rhombus in which the altitude from $D$ to side $AB$ bisects $AB$. Find the angles of the rhombus.

Show that the quadrilateral formed by joining the mid-points of the sides of a rhombus,taken in order,is a rectangle.