Some of the most scientifically valuable transiting planets are those that were already known from radial velocity (RV) surveys. This is primarily because their orbits are well characterized and they preferentially orbit bright stars that are the targets of RV surveys. The Transiting Exoplanet Survey Satellite ({it TESS}) provides an opportunity to survey most of the known exoplanet systems in a systematic fashion to detect possible transits of their planets. HD~136352 (Nu$^2$~Lupi) is a naked-eye ($V = 5.78$) G-type main-sequence star that was discovered to host three planets with orbital periods of 11.6, 27.6, and 108.1 days via RV monitoring with the HARPS spectrograph. We present the detection and characterization of transits for the two inner planets of the HD~136352 system, revealing radii of $1.482^{+0.058}_{-0.056}$~$R_oplus$ and $2.608^{+0.078}_{-0.077}$~$R_oplus$ for planets b and c, respectively. We combine new HARPS observations with RV data from Keck/HIRES and the AAT, along with {it TESS} photometry from Sector 12, to perform a complete analysis of the system parameters. The combined data analysis results in extracted bulk density values of $rho_b = 7.8^{+1.2}_{-1.1}$~gcm$^{-3}$ and $rho_c = 3.50^{+0.41}_{-0.36}$~gcm$^{-3}$ for planets b and c, respectively, thus placing them on either side of the radius valley. The combination of the multi-transiting planet system, the bright host star, and the diversity of planetary interiors and atmospheres means this will likely become a cornerstone system for atmospheric and orbital characterization of small worlds.
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