We study the normal-state electronic excitations probed by angle-resolved photoemission spectroscopy (ARPES) in ${\mathrm{Bi}}_{1.6}{\mathrm{Pb}}_{0.4}{\mathrm{Sr}}_2{\mathrm{CuO}}_6$ (Bi2201) and ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ (Bi2212). Our main goal is to establish explicit criteria for determining the Fermi surface from ARPES data on strongly interacting systems where sharply defined quasiparticles do not exist and the dispersion is very weak in parts of the Brillouin zone. Additional complications arise from strong matrix element variations within the zone. We present detailed results as a function of incident photon energy, and show simple experimental tests to distinguish between an intensity drop due to matrix element effects and spectral weight loss due to a Fermi crossing. We reiterate the use of polarization selection rules in disentangling the effect of umklapps due to the BiO superlattice in Bi2212. We conclude that, despite all the complications, the Fermi surface can be determined unambiguously; it is a single large hole barrel centered about $(\pi ,\pi )$ in both materials.