We consider a binary hypoeutectic alloy casting which solidifies in dendritic form in an unreinforced engineering casting and seek to predict its microstructure in a metal matrix composite. We focus on the case where the reinforcement is fixed in space and fairly homogeneously distributed. We assume that the reinforcement does not catalyze heterogeneous nucleation of the solid. We show that the reinforcement can cause several microstructural transitions in the matrix alloy, depending on the matrix cooling rate, the width, Lambda, of interstices left between reinforcing elements, and the initial velocity V of the solidification front. These transitions comprise the following: (1) coalescence of dendrite arms before solidification is complete, causing solidification to proceed in the later stages of solidification with a nondendritic primary phase mapping the geometry of interstices delineated by reinforcement elements; (2) sharp reduction or elimination of microsegregation in the matrix by diffusion in the primary solid matrix phase; and (3) a transition from dendrite to cell formation, these cells featuring significant undercoolings or a nearly plane front configuration when reinforcing elements are sufficiently fine. Quantitative criteria are derived for these transitions, based on previous work on composite solidification, observations from directional solidification experiments, and current solidification theory. Theory is compared with experimental data for aluminum-copper alloys reinforced with alumina fibers and for the dendrite to cell transition using data from directional succinonitrile-acetone solidification experiments. Theory and experiment show good agreement in both systems.