regular polygon

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 360^o circumference

of the circle is divided into 10 each parts, each 36^o.

 

The three sections have a total of 3 x 36^o  =  108^o.

 

The measure of the angle that you want is one-half of this intercepted arc or 54^o.

 

[There are other ways to solve this problem.]