$XYZW$ is a rhombus. If the diagonals $XZ$ and $YW$ intersect at $P$,then $\angle XPY = \ldots \ldots \ldots$ (in $^o$)

In parallelogram $ABCD$,$\angle A$ is an obtuse angle,$AM \perp BC$,and $AN \perp CD$. If $\angle MAN = 60^{\circ}$,find each of the angles of quadrilateral $ABCD$.

Can all the angles of a quadrilateral be right angles? Give reason for your answer.

$P$ 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.

$ABCD$ is a rectangle in which diagonal $BD$ bisects $\angle B$. Show that $ABCD$ is a square.

$E$ and $F$ are respectively the mid-points of the non-parallel sides $AD$ and $BC$ of a trapezium $ABCD$. Prove that $EF \parallel AB$ and $EF = \frac{1}{2}(AB + CD)$.