We investigate and discuss the origin of interface resistance in magnetic trilayers with the half-metallic Co${}_{2}$MnSi by performing first-principles electronic-structure and ballistic transport calculations for Co${}_{2}$MnSi/$X/$Co${}_{2}$MnSi(001) ($X=$ Ag, Au, Al, V, Cr). We found that the matching of the Fermi surface projected to the two-dimensional Brillouin zone of in-plane wave vector (${k}_{\ensuremath{\parallel}}$) is a main contributing factor for the spacer ($X$) dependence of the interfacial resistance. Furthermore, the MnSi-terminated interface shows low resistance compared with the Co-terminated interface because the Co-terminated interface has a larger $d$ component in the local density of states at the Fermi level than that of the MnSi-terminated interface. We conclude that Ag, Au, and Al spacers with MnSi termination of CMS$/X/$CMS trilayers will provide the large interfacial spin-asymmetry coefficient because of the small interface resistance in parallel magnetization.