If two sides of a triangle are given, how many triangles are possible? I thought myself that you could rotate one arbitrary side, which is connected to the other side, over 90 degrees to the left and 90 degrees to the right, so basically there are infinitely many triangles possible. 
Am I right? 

I agree with the other answers but would like to present a geometric argument which is very similar to the analysis already indicated by the OP.
Assume that two of the sides have a length of $x$ and $r$, where $x$ and $r$ are both positive.  Assume that one of the vertices of the triangle will have Cartesian coordinates $(x,0).$  Further assume that the origin [i.e. Cartesian coordinates $(0,0)$] will be a 2nd vertex of the triangle.
Now, consider a circle centered at the origin, of radius $r$.  You can select any point on this circle as the third vertex of the triangle [except for the points $(r,0)$ and $(-r,0)$].
