  New results on pentagonal geometries PENT(k,r) with block sizes k = 3 or k =
4 are given. In particular we completely determine the existence spectra for
PENT(3,r) systems with the maximum number of opposite line pairs as well as
those without any opposite line pairs. A wide-ranging result about PENT(3,r)
with any number of opposite line pairs is proved. We also determine the
existence spectrum of PENT(4,r) systems with eleven possible exceptions.
