Following earlier work by Aldo de Luca and others, we study trapezoidal words and their prefixes, with respect to their characteristic parameters K and R (length of shortest unrepeated suffix, and shortest length without right special factors, respectively), as well as their symmetric versions H and L. We consider the distinction between closed (i.e., periodic-like) and open prefixes, and between Sturmian and non-Sturmian ones. Our main results characterize right special and strictly bispecial trapezoidal words, as done by de Luca and Mignosi for Sturmian words.