The material properties of silicon nitride (SiN) play an important role in the performance of SiN membranes used in optomechanical applications. An optimum design of a subwavelength high-contrast grating requires accurate knowledge of the membrane thickness and index of refraction, and its performance is ultimately limited by material absorption. Here we describe a cavity-enhanced method to measure the thickness and complex index of refraction of dielectric membranes with small, but nonzero, absorption coefficients. By determining Brewster's angle and an angle at which reflection is minimized by means of destructive interference, both the real part of the index of refraction and the sample thickness can be measured. A comparison of the losses in the empty cavity and the cavity containing the dielectric sample provides a measurement of the absorption.