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An approach that allows one to determine the unloaded quality factor of a quasi-optical dielectric resonator (QDR) under conditions of the resonance line splitting corresponding to twofold degenerative whispering gallery (WG) mode has been proposed. The resonator is represented by an equivalent network as two oscillatory circuits with a certain coupling between them. The approach allows one to determine the Q-factor of a single oscillatory circuit using three measured parameters, namely: 1) ratio between specific amplitudes of the resonator amplitude-frequency response (or its second derivative); 2) resonant frequency; and 3) frequency splitting of the response (or its respective second derivative). The proposed technique is illustrated by the measurements in the millimeter-wave range for sapphire quasi-optical resonators with conducting and high-Tc superconducting (HTS) film endplates.