TITLE: Determine angle $x$ using only elementary geometry
QUESTION [9 upvotes]: Using only elementary geometry, determine angle x.

You may not use trigonometry, such as sines and cosines, the law of sines, the law of cosines, etc.

REPLY [7 votes]: Each node of the regular polygon $P_{18}$ sees all other nodes separated with $10^{\circ}$, making this $P_{18}$ a perfect drawing canvas for this problem (left image):

It is easy to verify that $A$ and $B$ are located on a large diagonal of $P_{18}$.
Now embed the unique, smaller $18$-gon $p_{18}$ inside $P_{18}$, with nodes $A$, $B$ and $C$.
Then we have:
$$AB \overset{\text{similarity}}{\parallel} A’B’\overset{\text{regularity}}{\parallel}A’’B’’$$
Now using the $10^{\circ}$-property, we see that: 
$$x = \angle A’’B’’B = 2\times 10^{\circ} = 20^{\circ} \qquad \blacksquare $$