Is my observation correct about geometric constructions?

Question

I have observed that it is possible to construct angles which are multiples of 3 with a ruler and a compass (Angles are in degrees). For example, 135°, 45° etc. can be constructed but Angles like 100° and 20° are impossible to draw with a ruler and compass. If my observation is correct, is there a proof of this statement?

Answer 1

Constructing an angle with an integer number of degrees not divisible by $3$ implies (and is equivalent to) the ability to construct a $1^\circ$ angle, from which one can construct a regular $9$-gon, and that cannot be done with ruler and compass.

Answer 2

We do the "multiples of $3^\circ$ are constructible" part.

Let $\theta$ be the $36^\circ$ angle. Then $\cos(3\theta)=-\cos(2\theta)$ and therefore $$4\cos^3\theta-3\cos\theta=1-2\cos^2\theta.$$ This has the root $\cos\theta=-1$, Dividing we find that $\cos\theta$ satisfies the equation $$4\cos^2\theta-2\cos\theta-1=0.$$ Thus $\cos(\theta)=\frac{1+\sqrt{5}}{4}$.

Since we can take square roots using straight edge and compass, it follows that the $36^\circ$ angle is constructible.

The $30^\circ$ angle is constructible. It follows that the $6^\circ$ angle is constructible, and therefore so is the $3^\circ$ angle, and therefore any integer multiple of $3^\circ$.

Answer 3

It is correct. $15^\circ$ is constructible as two bisections of $60^\circ$ and $27^\circ$ is constructible as two bisections of $108^\circ$ (the pentagon), so you can construct the GCD of these, which is $3^\circ$, so you can construct all multiples. Going the other way, one can construct a regular $17-$ gon, which implies the constructability of $\frac {360}{17}$ degrees, all multiples and all divisions by powers of $2$. One can also construct a $65537-$ gon, so there are many angles that are not multiples of $3^\circ$ that are constructible.