$O$ is any point inside the parallel lines $AB$ and $CD$. $OP$ and $OQ$ are two perpendiculars on $XY$ and $AB$ respectively. Let's prove that $P, O, Q$ are collinear.

Each pair of arms of two angles is parallel to each other. Let's prove either these angles are equal in measurement or they are supplementary.

The diagonal $AC$ of the parallelogram ABCD bisects $\angle BAD$. Let's prove that the diagonal AC also bisects $\angle BCD$.

Let's prove if one angle of a parallelogram is right angle, then all other angles of the parallelogram are also right angles.