In the Al-Co-Cu alloy system, both the decagonal quasicrystal with the space group of $Poverline{10}m2$ and its approximant Al$_{13}$Co$_4$ phase with monoclinic $Cm$ symmetry are present around 20 at.% Co-10 at.% Cu. In this study, we examined the crystallographic features of prepared Al-(30-x) at.% Co-x at.% Cu samples mainly by transmission electron microscopy in order to make clear the crystallographic relation between the decagonal quasicrystal and the monoclinic Al$_{13}$Co$_4$ structure. The results revealed a coexistence state consisting of decagonal quasicrystal and approximant Al$_{13}$Co$_4$ regions in Al-20 at.% Co-10 at.% Cu alloy samples. With the help of the coexistence state, the orientation relationship was established between the monoclinic Al$_{13}$Co$_4$ structure and the decagonal quasicrystal. In the determined relationship, the crystallographic axis in the quasicrystal was found to be parallel to the normal direction of the (010)$_{rm m}$ plane in the Al$_{13}$Co$_4$ structure, where the subscript m denotes the monoclinic system. Based on data obtained experimentally, the state stability of the decagonal quasicrystal was also examined in terms of the Hume-Rothery (HR) mechanism on the basis of the nearly-free-electron approximation. It was found that a model based on the HR mechanism could explain the crystallographic features such as electron diffraction patterns and atomic arrangements found in the decagonal quasicrystal. In other words, the HR mechanism is most likely appropriate for the stability of the decagonal quasicrystal in the Al-Co-Cu alloy system.
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