Definition: Let us refer to obtuse triangles with the largest angle strictly above a given cutoff value as 'strongly obtuse' - the definition is parametrized by the cutoff value. Likewise, strongly acute triangles are acute triangles with the smallest angle strictly less than a specified cutoff.
General Question: Given an n-gon, how does one triangulate it into a finite number of strongly obtuse/ strongly acute pieces with specified cutoff - resulting in least number of pieces? And before that, how to decide whether such a triangulation exists?
Example: it seems impossible to cut a square into finitely many strongly obtuse triangles with cutoff 120.
Guess: Consider partitioning a given n-gon into strongly obtuse triangles with some cutoff greater than or equal to 120 degrees. It appears that to check if the n-gon can be so triangulated, we need to see only its triangulations using only segments connecting the vertices of the input n-gon - adding extra vertices for the triangulation does not help. And for any cutoff less than 120 degrees, any triangle and hence, any n-gon can be easily cut into strongly obtuse triangles with extra vertices.
Note: For partition into strongly acute triangles, the 'first' case seems to be a cutoff of 36 degrees.