We present chemical abundances of red giant branch (RGB) stars in the dwarf spheroidal (dSph) satellite system of Andromeda (M31), using spectral synthesis of medium resolution (R $sim 6000$) spectra obtained with the Keck II telescope and DEIMOS spectrograph via the Spectroscopic and Photometric Landscape of Andromedas Stellar Halo (SPLASH) survey. We coadd stars according to their similarity in photometric metallicity or effective temperature to obtain a signal-to-noise ratio (S/N) high enough to measure average [Fe/H] and [$alpha$/Fe] abundances. We validate our method using high S/N spectra of RGB stars in Milky Way globular clusters as well as deep observations for a subset of the M31 dSphs in our sample. For this set of validation coadds, we compare the weighted average abundance of the individual stars with the abundance determined from the coadd. We present individual and coadded measurements of [Fe/H] and [$alpha$/Fe] for stars in ten M31 dSphs, including the first [$alpha$/Fe] measurements for And IX, XIV, XV, and XVIII. These fainter, less massive dSphs show declining [$alpha$/Fe] relative to [Fe/H], implying an extended star formation history. In addition, these dSphs also follow the same mass-metallicity relation found in other Local Group satellites. The conclusions we infer from coadded spectra agree with those from previous measurements in brighter M31 dSphs with individual abundance measurements, as well as conclusions from photometric studies. These abundances greatly increase the number of spectroscopic measurements of the chemical composition of M31s less massive dwarf satellites, which are crucial to understanding their star formation history and interaction with the M31 system.
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