In parallelogram $ABCD$,$\angle A = \angle B - 30^{\circ}$,then find $\angle A$. (in $^{\circ}$)

$P, Q, R$ 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.

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 $ABCD$.

The angle between two altitudes of a parallelogram drawn from the vertex of an obtuse angle is $60^{\circ}$. Find the angles of the parallelogram.

In $\Delta ABC$,$P$ and $Q$ are the midpoints of $AB$ and $AC$ respectively. If $BC + PQ = 21 \text{ cm}$,then find the length of $BC$.

$A$ square is inscribed in an isosceles right triangle so that the square and the triangle have one angle common. Show that the vertex of the square opposite the vertex of the common angle bisects the hypotenuse.