Find the angle between two lines using a compass and straight edge.

Question

I've drawn two random, non-parallel, straight lines on a plane. They cross over, forming two angles, $a$ and $b$, where ($a + b + a + b) = 1$ (or $360^\circ$) and $a ≤ b$. (Making $a$ either the acute angle or a right angle.)

Using only a compass and straight edge, how would I find the value of $a$? It is acceptable to use an infinite number of steps.

(This is an abstract recreational puzzle rather than a practical question. If it were practical, I'd acquire a protractor.)

Answer 1

Since you can bisect angles, erect a perpendicular to one of the lines, and use binary search to find the binary representation of how large a fraction of a right angle $a$ is.

Answer 2

Construct the angle $qa$ by making $q$ stacked copies of $a$. Count how many full turns you make in this process, say $p$. The value of $a$ is between $2\pi \frac p q$ and $2\pi\frac{p+1}q$. Now send $q$ to infinity.

(You did say "abstract" and "[im]practical"...)