What is the measure of the dent angle for a star-shaped polygon with 4 points and a point angle of 40°?
"Dent" angle? Do you mean the external angle between two successive points? Also, I will assume that this is a "regular" polygon with all point angles and sides equal.
Draw the lines connecting the bases of the points. The points form isosceles triangles with vertex angle of 40 degrees and so base angles of (180- 40)/2= 70 degrees. By symmetry, that forms a square which has angles of 90 degrees. The entire circle around such a point the is divided into four angels: the two 70 degree angles at the base of the two consectutive points, the right angle in the square, and the external angle. Calling the measure of the external angle $\displaystyle \theta$, we have $\displaystyle 70+ 70+ 90 +\theta= 230+ \theta= 360$ so $\displaystyle \theta= 360- 230= 130$ degrees.