We have determined the abundances of 16O, 17O, and 18O in 31 lunar samples from Apollo missions 11, 12, 15, 16, and 17 using a high-precision laser fluorination technique. All oxygen isotope compositions plot within +/-0.016 per mil (2 standard deviations) on a single mass-dependent fractionation line that is identical to the terrestrial fractionation line within uncertainties. This observation is consistent with the Giant Impact model, provided that the proto-Earth and the smaller impactor planet (named Theia) formed from an identical mix of components. The similarity between the proto-Earth and Theia is consistent with formation at about the same heliocentric distance. The three oxygen isotopes (Δ17O) provide no evidence that isotopic heterogeneity on the Moon was created by lunar impacts.