Euclidean construction of one degree angle

Question

We can construct 15 degrees by bisecting 60 degrees twice. We can construct 37 degrees by constructing a right angled triangle with sides 3 and 4 since tan 37 is 3÷4 We can also similarly construct 53 degrees When we have constructed 37 and 53 we can construct 53 -37 = 16 degrees Since we have constructed 15 and 16 we can construct 16-15 =1 degree But if we can construct 1 degree we can construct 20 degrees This means we can trisect 60 degrees which is supposed to be impossible by euclidean means What is the mistake in this procedure?

Answer 1

The mistake is that $\tan(37^\circ) \approx 0.7535540501 \neq 0.75 \approx \tan(36.87^\circ)$.

As an aside, it is not always necessarily impossible to trisect an angle - in a few special cases, there are some angles that can be trisected. For example, you can construct a $45^\circ$ angle by bisecting a right angle, and you can also construct a $15^\circ = \tfrac{1}{3}\cdot 45^\circ$ angle by bisecting a $30^\circ$ angle. So naturally, this means that a $45^\circ$ angle can be trisected.

The angle trisection constructability result you are referring to says rather that it is not possible to trisect all angles, in general. In particular, for angles of an integral amount of degrees (i.e. $1^\circ, 2^\circ, 3^\circ$, ...) it is possible to construct only the angles which are integer multiples of $3^\circ$ (i.e. $3^\circ, 6^\circ, 9^\circ$, ...).