A complete contact problem between elastically dissimilar materials is studied using an asymptotic analysis. A quarter plane wedge on a half-plane represents the contact edge geometry. Two eigenvalues are obtained for pairs of contacting materials, and their characteristics are classified on the Dundurs parallelogram. Generalized stress intensity factors, K_I and K_II, are derived to use a two-term stress equation of dimensionless form with developing a mode separation angle. It is found that the order of stress singularity increases as the wedge becomes more rigid than the half-plane. Slipping characteristics on the contact interface are investigated in detail, especially for the case of K_I < 0 < K_II that represents a typical adhesive complete contact condition. An example case is given using a finite element model to provide calibration of the stress intensities for a specific material, geometry and load combination.