I suspect a complete proof is not so easy. There is a field called "the constructible numbers," sometimes denoted $E$ for Euclid, which is the smallest subfield of the reals closed under square root of positive numbers. A length $x$ is constructible if $x \in E.$ An angle $\alpha$ is constructible if $\cos \alpha \in E,$ or $\sin \alpha \in E,$ or $\tan \alpha \in E,$ these conditions being equivalent. The constructible angles that are rational multiples of $\pi$ are known precisely. However, we get a mess as soon as we consider, say, $\arctan 2,$ which is a transcendental multiple of $\pi.$

Using roughly the symbols on your link, we want to show that, for **acute** constructible angles $\alpha, \beta$ and **positive** constructible lengths x,y, that
$$ \color{magenta}{ x \alpha + y \beta \; \notin \; E.} $$
Now, if $x, y \in \mathbb Z,$ this follows by Hermite-Lindemann, page 131, Theorem 9.11 (b) in Niven, *Irrational Numbers.* Diving by a different integer, it is also true if $x, y \in \mathbb Q.$ Finally, multiplying by a single irrational constructible number, it is also true if the ratio $x/ y \in \mathbb Q.$

Anyway, a proof of the proposition in color would finish this. Probably true, but...

Now that I compare the two, evidently Cebotarev and Dorodnow proved (I think) that the expression in color is only in the field $E$ if equal to $0,$ where they were concentrating on $x,y$ of opposite signs. So it would seem they had enough technology to finish this. I just don't know how it was done.

It seems Postnikov gave a summary of Cebotarev and Dorodnov in his book on Galois theory, an excerpt was translated in the MAA Monthly, and many of those columns collected in a book. see also Quadrature of the Lune

Hmmm...Postnikov evidently assumed commensurable angles which means he does not show how to deal with the problem above. At this point, and with my experience of Nestorovich and Mordukhai-Boltovskoi in about the same time, I have got to wonder if Cebotarev and his student really gave a complete proof of the five lunes, or made some "commensuability" assumption. As Samuel L. Jackson said in the Long Kiss Goodnight, "When you make an assumption, you make an ass out of you and umption." http://www.imdb.com/title/tt0116908/ and

Mitch Henessey: ...everyone knows, when you make an assumption, you make an ass out of "u" and "umption".

http://www.imdb.com/title/tt0116908/trivia?tab=qt&ref_=tt_trv_qu

NOTE: **it is supposed to be** When you assume, you make an "ass" out of "u" and "me."