TITLE: Is it possible to draw $85 ^{\circ} $ and $110 ^{\circ}$ angles by compass and straightedge construction?
QUESTION [6 upvotes]: Is it possible to draw $85 ^{\circ} $ and $110 ^{\circ}$ angles by compass and straightedge construction?If yes then how?

I thought about it and the only thing coming to mind other than a protractor is to use the concept of trigonometry functions but that still needs calculator,and the other ides coming to my mind involves trisecting an angle which is impossible,any other feasible ideas?

REPLY [3 votes]: It is a theorem of Gauss (and... Wantzel? Gauss did sufficient, someone else did necessary) that we can construct a regular $n$-gon via a straight edge and compass if and only if $n$ is of the form $$p_1p_2\ldots p_i2^j$$ where the $p_k$ are distinct Fermat Primes.
Now, if you can construct an angle of $d$-degrees then you can use this to construct a regular $n$-gon, where $n$ is such that $d.n=0 \text{ mod }360$ (draw your angle on a circle, follow where your lines hit the edge and repeat - basically, work out the order of $d$ in the cyclic group of order $360$).
If $d=110$ then $n=36=2^{2}3^{2}$.
If $d=85$ then $n=72=2^{3}3^{2}$.
Your Fermat Primes are $3$, $5$, $17$, $257$ and $65537$ (although I believe it is an open question if there are any more). However, they each must occur no more than once in the prime factorisation of $n$. $3$ appears twice in both $36$ and $72$ (aka $9$ divides them both), so we cannot construct a $36$-gon nor a $72$-gon. Thus, we cannot construct angles of either 110 or 85 degrees.