Each side of a convex pentagon is produced at both ends until it meets the productions of the two sides that are not adjacent to it, forming a five-pointed star. What is the sum of the five angles at the points of the star?

The sum is always 180°, whatever convex pentagon is used. One way to see it: each star point caps a triangle whose other two angles are exterior angles of the pentagon, so summing the five tip angles gives $5\times 180$° minus the total of ten such exterior angles. Those ten make up two full circuits of the pentagon's exterior angles, that is $2\times 360$° = 720°, and 900° − 720° = 180°. A regular pentagram checks it instantly: each tip measures 36° and $5\times 36$° = 180°. The trap 540° is the sum of the pentagon's INTERIOR angles, 360° is the sum of its exterior angles, and the answer is emphatically not shape-dependent.