The generalized equations of motion for the gyroresonance interaction between energetic electrons and coherent VLF waves in the magnetosphere are employed to study quasi-relativistic electron precipitation. It is suggested that these exact equations be used for particle energies higher than 50 keV (corresponding to a 10% correction in average pitch angle scattering with respect to a nonrelativistic formulation). Based on these equations and the relativistic cyclotron resonance condition, a previous test particle simulation method (Inan, 1977; Inan et al., 1978, 1982) is extended to higher energies and is used to study the wave-particle interactions involving the near-loss-cone quasi-relativistic particles. The root mean square (rms) pitch angle scattering of the near-loss-cone particles and the corresponding precipitated energy spectrum are then calculated and compared. It is argued that the full width at half maximum (FWHM) of the rms scattering pattern would give an upper bound to that of the energy spectrum of the precipitated flux. Furthermore the peak widths are demonstrated to be within the upper limit values measured by recent satellite experiments. This agreement between the computed and measured spectral width lends credence to the suggestions that the observed narrow peaks in precipitated energy spectra may indeed be caused by monochromatic signals injected into the magnetosphere by VLF transmitters.