#1**0 **

Because you have a reglular polygon, it can be inscribed in a circle.

The angle A_{1}A_{4}A_{8} cuts off the arc A_{8}A_{9}A_{10}A1.

Thre are three sections to this arc: A_{8}A_{9}, A_{9}A_{10}, and A_{10}A_{1}.

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.]

geno3141 Feb 22, 2021