**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.

maximumangle is less than some $\theta>60^\circ$. The condition of max angle greater than some threshold automatically implies obtuseness (for angles over $90^\circ$), but the condition of min angle under some threshold doesn't imply acuteness, so it's a little weird to label such a condition "strongly acute" when the property they satisfy is not actually stronger than acuteness. $\endgroup$both acute and isosceles( for example, Hoggatt, V. E. Jr. and Denman, R. "Acute Isosceles Dissection of an Obtuse Triangle." Amer. Math. Monthly 68, 912-913, 1961) appear to yield many acute triangles that are 'fat' and that motivated the present query on partitioning into thin acute and obtuse triangles. $\endgroup$3more comments