First of all, use correct terminology. To say that $\angle BPE=\angle DQE$ means we have one and the same point-set. From the diagram that is false.
What is true: $m(\angle BPE)=m(\angle DQE)$ the measures of those two angles are equal.
Now there are several additional pairs of congruent angles. You tell us what the are.
Are we to assume that line segments AB and CD are parallel? If they aren't then you can't be sure that any angles are equal in measure.
If they are, you need to look up "corresponding angles on parallel lines (the F rule)", "alternate angles on parallel lines (the Z rule)" and "vertically opposite angles (the X rule)".