Two forms of the so-called mixed-potential electric field integral equation (MPIE) are developed for two-dimensional, perfectly conducting (PC) surfaces of arbitrary shape in the presence of an infinite, PC wedge, subject to transverse electric excitation. One of the MPIEs is based on the Coulomb gauge while the other employs the Lorentz gauge. In either case the effect of the wedge is incorporated in the integral equation by means of the appropriate Green's functions, leaving the current distribution on the arbitrary surface as the only unknown. The Green's functions are derived by the eigenfunction expansion technique. A well-established moment method procedure is adapted to numerically solve both forms of the MPIE. Computed results are presented for several cases of interest, and the relative merits of the Coulomb and Lorentz gauge MPIEs are discussed.