||Let Delta be a hyperbolic triangle with a fixed area phi. We prove that for all but countably many phi, generic choices of Delta have the property that the group generated by the pi-rotations about the midpoints of the sides of the triangle admits no nontrivial relations. By contrast, we show for all phi is an element of (0, pi)\Q pi, a dense set of triangles does afford nontrivial relations, which in the generic case map to hyperbolic translations. To establish this fact, we study the deformation space C-theta of singular hyperbolic metrics on a torus with a single cone point of angle theta = 2(pi - phi), and answer an analogous question for the holonomy map rho(xi) of such a hyperbolic structure xi. In an appendix by Gao, concrete examples of theta and xi is an element of C-theta are given where the image of each rho(xi) is finitely presented, non-free and torsion-free; in fact, those images will be isomorphic to the fundamental groups of closed hyperbolic 3-manifolds.