The detailed chemical abundances and atmospheric parameters of 27 K-type giantstars in the Sagittarius Dwarf Spheroidal galaxy (Sgr dSph) and a control sample of27 K-type giant stars in the Galactic Bulge have been calculated. The the chemicalcomposition of Sgr dSph stars are different from the Galactic Bulge stars. The SgrdSph stars are metal poor, with a mean metallicity of [Fe/H] = -0.82 dex with arange of -1.31 to -0.35 dex and an [alpha/Fe] = 0.20 dex with range of -0.28 to 0.55dex. For the Bulge a mean metallicity was found of [Fe/H] = -0.25 dex with a rangeof -1.15 to +0.53 dex and a mean [alpha/Fe] = 0.06 dex with range of -0.41 to 0.48 dex.Many elements (Si, Ca, Ti, Fe) in the Sgr dSph show an under-abundance whencompared to the Bulge stars. The sample of stars studied in both galaxies does notinclude metal-rich populations which are seen in other studies because the selectioncriteria are biased towards metal-poor stars, which affects the mean metallicity andrange found in this project.For the Sgr dSph, the [alpha/Fe] ratio declines with [Fe/H] from 0.4 to -0.1 dex and the[alpha/Fe] knee is at approximately -0.8 dex. For the Bulge, the [alpha/Fe] ratio declineswith [Fe/H] from 0.4 dex to -0.3 dex and the [alpha/Fe] knee is at approximately -0.6 dex. The difference in the metallicity of the [alpha/Fe] knees support the fact theenrichment occurred faster in the Bulge stars. Additionally, the low [alpha/Fe] ratiosimply that type Ia supernovae contributed to the composition of the younger, metalrich Sgr dSph population. The extent to which the Sgr dSph has changed by theinteraction with the Milky Way is currently difficult to assess. The lack of radialabundance variations in the Sgr dSph show that the population is homogeneous and13 all the populations are seen at all radii. This is different to other dwarf galaxies andis likely a result of tidal stripping that is going on in the Sgr dSph.