I'm going to make an interpretation of the question, and then answer it. If my interpretation is wrong, OP can let us know.
We are given triangles $T_1,T_2,\dots,T_n$, and we want to know whether it is possible to tessellate an arbitrary polygonal region $P$ with a finite number of triangles, each triangle similar to one of those given.
I claim it's not possible. Let $P$ have an angle that is smaller than any of the angles in the triangles. Then there is no way to get to that angle.
Now, what if we are allowed to pick the triangles $T_1,T_2,\dots,T_n$ after we have seen the region $P$? If $n$ is fixed, we're still out of luck. The angles we can get lie in an extension field of transcendence degree at most $2n+1$ over the rationals, so if we are faced with a region with more than $2n+1$ algebraically independent angles, we can't tessellate it.
In short, under the sort of assumptions I've been making, the class of tessellatable polygonal regions is very restricted.