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In models with a U(1) gauge extension beyond the Standard Model, one canderive sum rules for the couplings of the theory that are a consequence oftree-level unitarity. In this paper, we provide a comprehensive list ofcoupling sum rules for a general SU(2)$_L\times$U(1)$_Y \times$U(1)$_{Y'}$gauge theory coupled to an arbitrary set of fermion and scalar multiplets.These results are of particular interest for models of dark matter that employan extended gauge sector mediated by a new (dark) $Z^\prime$ gauge boson. Forthe case of a minimal extension of the Standard Model with a U(1)$_{Y'}$ gaugeboson, we clarify the definitions of the weak mixing angle and the electroweak$\rho$ parameter. We demonstrate the utility of a generalized $\rho$ parameter(denoted by $\rho^\prime$) whose definition naturally follows from theunitarity sum rules developed in this paper.

Tree-level Unitarity in SU(2)$_L\times$U(1)$_Y \times$U(1)$_{Y'}$ Models