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Consider a 3$-$dimensional manifold $N$ obtained by gluing a finite number ofideal hyperbolic tetrahedra via isometries along their faces. By varying theisometry type of each tetrahedron but keeping fixed the gluing pattern wedefine a space $\mathcal{T}$ of complete hyperbolic metrics on $N$ with conesingularities along the edges of the tetrahedra. We prove that $\mathcal{T}$ ishomeomorphic to a Euclidean space and we compute its dimension. By means ofexamples, we examine if the elements of $% \mathcal{T}$ are uniquely determinedby the angles around the edges of $N.$

Generalized Teichm\"{u}ller space of non-compact 3-manifolds and Mostow rigidity