While studying a course on field theory, my prof gave the definition of constructible angle as
An angle $\theta (0\leq\theta\leq2\pi)$ is said to be constructible, iff the length $\cos\theta$ is constructible.
After a some search in textbooks and reference books (and also on the internet), I found one more definition:
An angle $\theta$ is said to be constructible if it is possible to construct two intersecting lines that form an angle of $\theta$
However for the the following definition looks more natural, but surprisingly it doesnt appear in any standard textbook
Edit
An angle $\theta$ is said to be constructible, if it can be constructed using ruler and compass.
Is there something wrong with this definition? If so, can someone explain?
-- Mike
Contrary to some of the comments, I would not call the proposed "definition" of a constructible angle circular, but I would say it is useless for its avowed purpose.
When we use the word "constructible," we do need to say more specifically what we mean by it, since the choice of tools determines what we can construct. (Even the ancient Greeks classified constructions by which tools you used.) But once we decide we want compass-and-straightedge constructions, we say, "An object is said to be constructible if it can be constructed using straightedge and compass." It is now trivial to substitute "an angle $\theta$" for the more general "an object," but until we say what it means to construct an angle with ruler and compass, the definition is of little use.
Hence we see textbooks explain "constructible angle" using a much more specific description of exactly what the results of such a construction would be. (The two definitions given describe different constructed objects but can be seen to be equivalent, as you can easily get the object described in either definition from the object described in the other.)