Let A_1 A_2 A_3 ... A_10 be a regular polygon. Find angle A_1 A_4 A_8.
Because you have a reglular polygon, it can be inscribed in a circle.
The angle A1A4A8 cuts off the arc A8A9A10A1.
Thre are three sections to this arc: A8A9, A9A10, and A10A1.
Since each portion of this arc is each to every other portion and there are 10 portions in the polygon, the 360o circumference
of the circle is divided into 10 each parts, each 36o.
The three sections have a total of 3 x 36o = 108o.
The measure of the angle that you want is one-half of this intercepted arc or 54o.
[There are other ways to solve this problem.]