Following Partition of polygons into 'strongly acute' and 'strongly obtuse' triangles, we record another triangulation question.
Question: Given an n-vertex polygonal region ("n-gon") and a real number $\theta$ < $\pi$, is it always possible to partition the n-gon into finitely many triangles such that each triangle has one angle equal to $\theta$? If the answer is "not always" (which appears to be the case), deciding whether such a triangulation exists for a given polygon and $\theta$ becomes another question. And in the cases where such a triangulation is possible, we could try to minimize the number of triangles.
An earlier post from the same ballpark: Cutting convex polygons into triangles of same diameter