Properties of a Rectangle

A transversal cuts two parallel lines at $A$ and $B$. The two interior angles at $A$ are bisected and so are the two interior angles at $B$; the four bisectors from a quadrilateral $ACBD$. Prove that: $\mathrm{CD}$ is parallel to the original parallel lines.

Calculate the angle marked with small letters in the diagram drawn.

A transversal cuts two parallel lines at $A$ and $B$. The two interior angles at $A$ are bisected and so are the two interior angles at $B$; the four bisectors from a quadrilateral $ACBD$. Prove that: $ACBD$ is a rectangle.

Prove that in a rectangle, the diagonals are of equal lengths.

The diagonals $AC$ and $BD$ of a rectangle $ABCD$ intersect each other at $P.$ If $\angle ABD=50°,$ and $\angle DPC=k°$ then find $k$. 

Calculate the angle marked with small letter in the diagram drawn.