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We report the detection of 114 extreme ultraviolet (EUV; 58 - 740 A) sources, of which 99 are new serendipitous sources, based on observations made with the imaging telescopes on board the Extreme Ultraviolet Explorer (EUVE) during the Right Angle Program (RAP). These data were obtained using the survey scanners and the Deep Survey instrument during the first year of the spectroscopic guest observer phase of the mission, from January 1993 to January 1994. The data set consists of 162 discrete pointings whose exposure times are typically two orders of magnitude longer than the average exposure times during the EUVE all-sky survey. Based on these results, we can expect that EUVE will serendipitously detect approximately 100 new EUV sources per year, or about one new EUV source per 10 sq deg, during the guest observer phase of the EUVE mission. New EUVE sources of note include one B star and three extragalactic objects. The B star (HR 2875, EUVE J0729 - 38.7) is detected in both the Lexan/B (approximately 100 A) and Al/Ti/C (approximately 200 A) bandpasses, and the detection is shown not to be a result of UV leaks. We suggest that we are detecting EUV and/or soft x rays from a companion to the B star. Three sources, EUVE J2132+10.1, EUVE J2343-14.9, and EUVE J2359-30.6 are identified as the active galactic nuclei MKN 1513, MS2340.9-1511, and 1H2354-315, respectively.