What is the sum of the measures of angles A, B, C, D, and E?
I really need at least some kind of hint here, because I have no idea where to begin. No information is given for the picture.
I have added five letters to your diagram.
The sum of the angles in the pentagon, KJLMN, is $\displaystyle 3\pi =540^\circ $. WHY?
In $\displaystyle \Delta\text{ANK}$ the sum $\displaystyle \angle A + \angle 1 + \angle 2 = 180^ \circ$.
Moreover, by the external angle theorem $\displaystyle \angle K = \angle A + \angle 1$.
If you do this for each of the other four “external triangles” and all five you will see the answer.
When you move a single vertex (at least if it remains on the same side of the line segment joining the oposite vertices) the angle sum remains invariant.
Since the given figure can be transformed by a sequence of such moves into a regular pentagram, the angle sum of the regular pentagram is equal to the angle sum of the given figure.
CB