Pedantry: it doesn’t have to be flat - for example a triangulated parabola could also have that configuration. You only get a topological result from just knowing edge and vertex counts. Now if the triangles are identical equilateral then you’re in business.
What if the triangles are all congruent but not equilateral? Can that even happen? That’s a fun one, so I won’t spoil it.
What if the triangles are all congruent but not equilateral? Can that even happen? That’s a fun one, so I won’t spoil it.