How does one solve arbitrary polygons, in the same sense as one solves a triangle?

Let us say you are given a polygon, and also are given some, but not all, of its angle measures and side lengths. How would one compute the following:

1. If there is a finite number (zero inclusive) of polygons described by the givens, then list them.
2. If there are an infinite number of polygons, then say so.

I was thinking that you could break the polygon into a triangle for each triplet of points, and start solving them, but I was thinking that this would be inefficient, and may not solve $1$ and $2$.

Is there any literature or algorithms relating to this problem, or am I on my own?

Note: It is a bonus if the algorithm also calculates the area during the calculation, but I know methods to do so if you have all the sides and angles.

• Intuitively I would expect there to be an algorithm like: 1) if there are any three pieces of data in a row (either angle-side-angle or side-angle-side), you can use them to eliminate a side (possibly multiplying the number of possibilities somewhat); 2) If you've done that as many times as possible and you're still not down to a triangle, there are an infinite number of possibilities. But the case analysis to prove this seems tedious... – Micah Mar 26 '14 at 2:54