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This problem trains for: SAT-II, AMC-8, AMC-10.

What is the angle between the shortest and the next to shortest adjacent diagonals of a regular n-gon?

Pick one of the shortest diagonals and the next shortest one adjacent to it. In the figure, consider the angle x between the diagonals AC and AD:

figure

There is another set of adjacent diagonals with the same lengths, AC and AM:

figure

Therefore, the problem as stated has two possible answers.

Let us find the measure of the angle x.

For the regular n-gon, the measure of an interior angle ∠a is:

equation

The triangle CBA is isosceles because the n-gon is regular and all edges have the same length. We can compute measure of the angle ∠CAB:

equation

Since ABCD is an isosceles trapezoid, we can find the measure of the angle ∠DAB:

equation

x is the difference:

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The measure of the angle y can be computed similarly.