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One of the fundamental structural properties of many networks is triangle closure. Whereas the influence of this transitivity on a variety of contagion dynamics has been previously explored, existing models of coevolving or adaptive network systems typically use rewiring rules that randomize away this important property, raising questions about their applicability. In contrast, we study here a modified coevolving voter model dynamics that explicitly reinforces and maintains such clustering. Carrying out numerical simulations for a variety of parameter settings, we establish that the transitions and dynamical states observed in coevolving voter model networks without clustering are altered by reinforcing transitivity in the model. We then use a semi-analytical framework in terms of approximate master equations to predict the dynamical behaviors of the model for a variety of parameter settings.