As part of a larger project to complete a comprehensive catalog of astrophysically relevant emission lines in support of new-generation X-ray observatories using the Lawrence Livermore electron beam ion traps EBIT-I and EBIT-II, we present observations of argon lines in the extreme-ultraviolet region. Our database includes wavelength measurements with standard errors, relative intensities, and line assignments for Ar IX-Ar XVI between 20 and 50 Å. The experimental data are complemented with a full set of calculations using the Hebrew University Lawrence Livermore Atomic Code (HULLAC). Despite differences in calculated and measured wavelengths, we find the calculated lines to be of great utility in analyzing our laboratory spectra. The calculated line intensities are generally sufficient to identify the strongest transitions in each charge state. We note, however, an underestimation by theory of the strength of the 3s-->2p lines relative to the 3d-->2p lines in Ar IX, Ar X, and Ar XI. The laboratory data are compared with Chandra observations of Procyon, resulting in the identification of an Ar IX line that was previously thought to be from S IX.