Abstract: We are interested in generalizing the classical Cauchy Rigidity Theorem and the Aleksandrov's Existence Theorem for convex polyhedra and the sphere to the closed surfaces of constant negative curvature. We prove that tessellations of the surfaces with hyperbolic metric cannot be infinitesimally perturbed so as to preserve their face angles. This is an analog of the Cauchy Rigidity for "polyhedra" on the hyperbolic surfaces. We also show an analog of the Cauchy Rigidity Theorem for tessellations in the same isotopy class of the tessellations. For existence we show that in a certain "convex hull" construction any set of cone angles can be realized which provides a first step in the suggested proof of the analog of the Aleksandrov's Theorem.