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The peculiarities of the interaction of the transition radiation of a charged particle with periodically modulated in space anisotropic magnetodielectric medium in the waveguide in the frequency region of the strong (resonant) interaction of the radiation with the modulated medium are considered. It is assumed that the charged particle moves with constant velocity perpendicular to the axis of the waveguide. Analytical expressions are obtained for the energy of transition radiation of transverse-electric (TE) and transverse-magnetic (TM) fields in the waveguide in the region of strong interaction with an accuracy of small modulation indexes in the first degree inclusive. Analytical expressions for the frequency and for the width of the frequency region of strong interaction are found. The possibility of the appearance of Cherenkov radiation in the region of strong interaction in the case of a rectangular waveguide is analyzed.