The response of a material to external stimuli depends on its low-energy excitations. In conventional metals, these excitations are electrons on the Fermi surface—a contour in momentum (k) space that encloses all of the occupied states for non-interacting electrons. The pseudogap phase in the copper oxide superconductors, however, is a most unusual state of matter1. It is metallic, but part of its Fermi surface is 'gapped out' (refs 2, 3); low-energy electronic excitations occupy disconnected segments known as Fermi arcs4. Two main interpretations of its origin have been proposed: either the pseudogap is a precursor to superconductivity5, or it arises from another order competing with superconductivity6. Using angle-resolved photoemission spectroscopy, we show that the anisotropy of the pseudogap in k-space and the resulting arcs depend only on the ratio T/T*(x), where T*(x) is the temperature below which the pseudogap first develops at a given hole doping x. The arcs collapse linearly with T/T*(x) and extrapolate to zero extent as T0. This suggests that the T=0 pseudogap state is a nodal liquid—a strange metallic state whose gapless excitations exist only at points in k-space, just as in a d-wave superconducting state.