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.