how can you tell if an angle is possible to construct? How can someone tell if an angle is possible to construct using a compass and unmarked ruler?  Is there any definite way to say if it is possible or not and construct it?
 A: There is a subfield of the real numbers, called the constructible numbers; call it $E$ for Euclidean. $E$ is the smallest ordered field such that, whenever $w \in E$ and $w > 0,$ we also have $\sqrt w \in E.$  
An angle $\theta$ is constructible if and only if
$$ \sin \theta \in E, $$ or
$$ \cos \theta \in E, $$ or
$$ \tan \theta \in E, $$ 
these three conditions being equivalent. 
The part that is surprising is that the constructible angles on the surface of the unit sphere, or the hyperbolic plane with curvature $-1,$ are exactly the same. The proof of all this goes back to the 1930's or so, Mordukhai-Boltovskoi and later, Nestorovich. See my article on constructions in the hyperbolic plane.
A: Only certain angles are possible.
Any angle can be bisected so angles like 180,90,45,22.5,etc are easy. As are half of other possible angles.
So lets focus on angles which are an odd fraction of a circle. E.g. 120 is a third of a circle - I imagine you know how to construct this one using radii of a circle to make a hexagon.
Gauss postulated (and was proven by Pierre Wantzel) that a n-sided polygon can be construct with straightedge and compass if the odd prime factors of n are distinct Fermat primes. So 5, 17, 257, etc are possible.
The first few constructible regular polygons are (from which you can calculate the corresponding internal or external angle):
3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, 257, 272... (sequence A003401 in OEIS)
Further reading: https://en.wikipedia.org/wiki/Compass-and-straightedge_construction#Constructible_angles
