Plane graphs with straight edges whose bounded faces are acute triangles

Let Tn denote a graph obtained as a triangulation of an n-gon in the plane. A cycle of Tn is called an enclosing cycle if at least one vertex lies inside the cycle. It is proved that a Tn admits a straight-line embedding in the plane whose bounded faces are all acute triangles if and only if Tn has no enclosing cycle of length ⩽4. Those Tn that admit straight-line embeddings in the plane without obtuse triangle are also characterized.