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In this paper we present observations, simulations, and analysis demonstrating the direct connection between the location of foreground emission on the sky and its location in cosmological power spectra from interferometric redshifted 21 cm experiments. We begin with a heuristic formalism for understanding the mapping of sky coordinates into the cylindrically averaged power spectra measurements used by 21 cm experiments, with a focus on the effects of the instrument beam response and the associated sidelobes. We then demonstrate this mapping by analyzing power spectra with both simulated and observed data from the Murchison Widefield Array. We find that removing a foreground model that includes sources in both the main field of view and the first sidelobes reduces the contamination in high k∥ modes by several per cent relative to a model that only includes sources in the main field of view, with the completeness of the foreground model setting the principal limitation on the amount of power removed. While small, a percent-level amount of foreground power is in itself more than enough to prevent recovery of any Epoch of Reionization signal from these modes. This result demonstrates that foreground subtraction for redshifted 21 cm experiments is truly a wide-field problem, and algorithms and simulations must extend beyond the instrument's main field of view to potentially recover the full 21 cm power spectrum. ; J.C.P. is supported by an NSF Astronomy and Astrophysics Fellowship under award AST-1302774. This scientific work makes use of the Murchison Radio-astronomy Observatory, operated by CSIRO. We acknowledge the Wajarri Yamatji people as the traditional owners of the Observatory site. Support for the MWA comes from the U.S. National Science Foundation (grants AST-0457585, PHY-0835713, CAREER-0847753, and AST-0908884), the Australian Research Council (LIEF grants LE0775621 and LE0882938), the U.S. Air Force Office of Scientific Research (grant FA9550-0510247), and the Centre for All-sky Astrophysics (an Australian Research Council Centre of Excellence funded by grant CE110001020). Support is also provided by the Smithsonian Astrophysical Observatory, the MIT School of Science, the Raman Research Institute, the Australian National University, and the Victoria University of Wellington (via grant MED-E1799 from the New Zealand Ministry of Economic Development and an IBM Shared University Research Grant). The Australian Federal government provides additional support via the Commonwealth Scientific and Industrial Research Organization (CSIRO), National Collaborative Research Infrastructure Strategy, Education Investment Fund, and the Australia India Strategic Research Fund, and Astronomy Australia Limited, under contract to Curtin University.