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We present an analysis of the eclipsing single-lined spectroscopic binary system a Dra based on photometry from the Transiting Exoplanet Survey Satellite (TESS) mission and newly acquired spectroscopic measurements. Recently discovered to have eclipses in the TESS data, at a magnitude of V = 3.7, alpha Dra is now one of the brightest detached eclipsing binary (EB) systems known. We obtain the parameters of this system by simultaneously fitting the TESS light curve in conjunction with radial velocities (RVs) acquired from the SONG spectrograph. We determine the fractional radii (Rla) for the primary and secondary components of the system to be 0.0479 +/- 0.0003 and 0.0226 +/- 0.0005, respectively. We constrain the temperature, mass, and luminosity (log(L/L-circle dot)) of the primary to be 9975 +/- 125 K, 3.7 +/- 0.1 M-circle dot, and 2.49 +/- 0.02, respectively, using isochrone fitting. Although the secondary is too faint to appear in the spectra, the obtained mass function and observed inclination yields a secondary minimum mass of M-2 = 2.5 +/- 0.1 M-circle dot, which suggests that it is an A2V type star. We were unable to obtain RVs of the secondary, and are only able to see a weak highly rotationally broadened absorption line, indicating that the secondary is rapidly rotating (vsin i similar to 200 km s(-1)). We also perform an abundance analysis of the primary star for 21 chemical elements. We find a complex abundance pattern, with a few elements having mild underabundances while the majority have solar abundances. We make available the PYTHON code used in this paper to facilitate future modelling of EBs. https://github.com/danhey/adra