  Understanding the composition and structure of the broad-line region (BLR) of
active galactic nuclei (AGN) is important for answering many outstanding
questions in supermassive black hole evolution, galaxy evolution, and
ionization of the intergalactic medium. We used single-epoch UV spectra from
the Cosmic Origins Spectrograph (COS) on the Hubble Space Telescope to measure
EUV emission-line fluxes from four individual AGN with $0.49 \le z \le 0.64$,
two AGN with $0.32 \le z \le 0.40$, and a composite of 159 AGN. With the Cloudy
photoionization code, we calculated emission-line fluxes from BLR clouds with a
range of density, hydrogen ionizing flux and incident continuum spectral
indices. The photoionization grids were fit to the observations using
single-component and locally optimally emitting cloud (LOC) models. The LOC
models provide good fits to the measured fluxes, while the single-component
models do not. The UV spectral indices preferred by our LOC models are
consistent with those measured from COS spectra. EUV emission lines such as N
IV \lambda 765, O II \lambda 833, and O III \lambda 834 originate primarily
from gas with electron temperatures between 37000 K and 55000 K. This gas is
found in BLR clouds with high hydrogen densities (n_H \ge 10^12 cm^-3) and
hydrogen ionizing photon fluxes (\Phi_H \ge 10^22 cm^-2 s^-1).
