We use resonant elastic x-ray scattering to determine the evolution of magnetic order in EuCd$_2$As$_2$ below $T_textrm{N}=9.5$,K, as a function of temperature and applied magnetic field. We find an A-type antiferromagneticstructure with in-plane magnetic moments, and observe dramatic magnetoresistive effects associated with field-induced changes in the magnetic structure and domain populations. Our textit{ab initio} electronic structure calculations indicate that the Dirac dispersion found in the nonmagnetic Dirac semimetal Cd$_3$As$_2$ is also present in EuCd$_2$As$_2$, but is gapped for $T < T_textrm{N}$ due to the breaking of $C_3$ symmetry by the magnetic structure.
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