We study TEM-wave propagation in a parallel plate waveguide with a graded transition between a lossy right-handed material (RHM) and an impedance-matched lossy left-handed material (LHM). The transition between the two media is graded along the direction perpendicular to the boundary between the two materials, and the permittivity and permeability vary according to hyperbolic tangent functions. We obtain exact analytical solutions to Maxwell's equations for lossy media, and the solutions for the field components confirm the expected properties of RHM-LHM structures. Finally, we perform a numerical study of the wave propagation over an impedance-matched graded RHM-LHM interface, using COMSOL, and obtain an excellent agreement between the numerical simulations and analytical results.