  Let M be a complete finite-volume hyperbolic 3-manifold with compact
non-empty geodesic boundary and k toric cusps, and let T be a geometric
partially truncated triangulation of M. We show that the variety of solutions
of consistency equations for T is a smooth manifold or real dimension 2k near
the point representing the unique complete structure on M. As a consequence,
the relation between deformations of triangulations and deformations of
representations is completely understood, at least in a neighbourhood of the
complete structure. This allows us to prove, for example, that small
deformations of the complete triangulation affect the compact tetrahedra and
the hyperbolic structure on the geodesic boundary only at the second order.
