In this paper we study the resonant coupling of fast magnetosonic waves to the Alfven or slow magnetosonic waves in general geometries. If the magnetic field lines are straight and the Alfven velocity does not vary along them, the linearized magnetohydrodynamic (MHD) equations exhibit a singularity at a given line which manifests the resonant coupling on that line. However, if the lines are not straight, or if the Alfven velocity does vary along them (or both) no singularity in the linearized MHD equations is apparent. Nonetheless, we show here that the solutions of the linearized MHD equations behave singularly in the neighborhood of an entire field line. This gives rise to the resonant coupling of the fast to the Alfven or slow waves along the entire line. We calculate the solutions near the resonant magnetic field line as well as the nonzero net energy flux across it. The method used is applicable to general geometries and plasma pressures.