The angle between two altitudes of a parallelogram through the vertex of an obtuse angle of the parallelogram is $60°.$ Find the angles of the parallelogram.

In the adjoining figure, $ABCD$ is a parallelogram in which $AB$ is produced to $E$ so that $BE=AB$. Prove that $ED$ bisects $BC$.

In the adjoining figure, $ABCD$ is a parallelogram and $E$ is the midpoint of side $BC$. If $DE$ and $AB$ when produced to meet at $F,$ prove that $AF=2AB$.

Two parallel lines $l$ and $\mathrm{m}$ are intersected by a transversal $t$. Show that the quadrilateral formed by the bisectors of interior angles is a rectangle.

A $∆\mathrm{ABC}$ is given. If lines are drawn through $\mathrm{A},\mathrm{B},\mathrm{C},$ parallel respectively to the sides $\mathrm{BC}, \mathrm{CA}$ and $AB$, forming $\Delta PQR$, as shown in the adjoining figure, show that $BC=\frac{1}{2}QR$.

In the adjoining figure, $△ABC$ is a triangle and through $A,B,C$ lines are drawn, parallel respectively to $BC, CA$ and $AB,$intersecting at $\mathrm{P}$, $Q$ and $R$. Prove that the perimeter of $△PQR$ is double the perimeter of $△ABC$.