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We study the geometry of the active galactic nucleus (AGN) obscurer in IRAS 09104+4109, an IR-luminous, radio-intermediate FR-I source at z = 0.442, using infrared data from Spitzer and Herschel, X-ray data from NuSTAR, Swift, Suzaku, and Chandra, and an optical spectrum from Palomar. The infrared data imply a total rest-frame 1-1000 micron luminosity of 5.5 × 10(exp 46) ergs/s and require both an AGN torus and a starburst model. The AGN torus has an anisotropy-corrected IR luminosity of 4.9 × 10(exp 46) ergs/s and a viewing angle and half-opening angle both of approximately 36deg from pole-on. The starburst has a star formation rate of (110 +/- 34) Stellar Mass/yr and an age of <50 Myr. These results are consistent with two epochs of luminous activity in IRAS 09104+4109: one approximately 150 Myr ago, and one ongoing. The X-ray data suggest a photon index of Gamma approx. =l 1.8 and a line-of-sight column density of N(sub H) approx. = 5 × 10(exp 23) sq cm. This argues against a reflection-dominated hard X-ray spectrum, which would have implied a much higher N(sub H) and luminosity. The X-ray and infrared data are consistent with a bolometric AGN luminosity of L(sub bol) approx.(0.5-2.5) ×10(exp 47) ergs/s. The X-ray and infrared data are further consistent with co-aligned AGN obscurers in which the line of sight "skims" the torus. This is also consistent with the optical spectra, which show both coronal iron lines and broad lines in polarized but not direct light. Combining constraints from the X-ray, optical, and infrared data suggest that the AGN obscurer is within a vertical height of 20 pc, and a radius of 125 pc, of the nucleus.