The nonlinear properties of ion-acoustic (IA) waves are investigated in a relativistically degenerate magnetized quantum plasma, whose constituents are non-degenerate inertial ions, degenerate electrons and immobile positively-charged heavy elements. For nonlinear studies, the well-known reductive perturbation technique is employed to derive the Korteweg-de Vries-Burger equation in the presence of relativistically degenerate electrons. Numerically, the amplitude, width, and phase speed are shown to be associated with the localized IA solitons, and shocks are shown to be significantly influenced by the various intrinsic parameters relevant to our model. The solitary and the shock wave properties have been to be influenced in the non-relativistic, as well as the ultrarelativistic, limits. The effects of the external magnetic field and the obliqueness are found to change the basic properties of IA waves significantly. The present analysis can be useful in understanding the collective process in dense astrophysical environments, like there of non-rotating white dwarfs, neutron stars, etc.