1958 AMC 12 Problem 15

Let the four arcs cut off by the quadrilateral’s sides have measures summing to $360^\circ$

An angle in the outside segment of a side subtends the complementary major arc

Let the four minor arcs cut off by the sides have measures $\alpha_1,\ldots,\alpha_4,$ whose sum is $360^\circ.$ The angle in the outside segment corresponding to arc $\alpha_i$ subtends the other arc, so its measure is $$\frac{360^\circ-\alpha_i}{2}.$$ The sum of the four angles is therefore $$\frac{4(360^\circ)-360^\circ}{2}=540^\circ.$$

Thus, the correct answer is D.