We present a determination of the precise fundamental physical parameters of 20 detached, double-lined, eclipsing binary stars in the Large Magellanic Cloud (LMC) containing G- or early K-type giant stars. Eleven are new systems; the remaining nine are systems already analyzed by our team for which we present updated parameters. The catalog results from our long-term survey of eclipsing binaries in the Magellanic Clouds suitable for high-precision determination of distances (the Araucaria Project). The V-band brightnesses of the systems range from 15.4 to 17.7mag, and their orbital periods range from 49 to 773days. Six systems have favorable geometry showing total eclipses. The absolute dimensions of all eclipsing binary components are calculated with a precision of better than 3%, and all systems are suitable for a precise distance determination. The measured stellar masses are in the range 1.4 to 4.6M_{sun}_, and comparison with the MESA isochrones gives ages between 0.1 and 2.1Gyr. The systems show an age-metallicity relation with no evolution of metallicity for systems older than 0.6Gyr, followed by a rise to a metallicity maximum at age 0.5Gyr and then a slow metallicity decrease until 0.1Gyr. Two systems have components with very different masses: OGLE LMC-ECL-05430 and OGLE LMC-ECL-18365. Neither system can be fitted by a single stellar evolution isochrone, explained by a past mass transfer scenario in the case of ECL-18365 and a gravitational capture or hierarchical binary merger scenario in the case of ECL-05430. The longest-period system, OGLE LMC SC9_230659, shows a surprising apsidal motion that shifts the apparent position of the eclipses. This is a clear sign of a physical companion to the system; however, neither investigation of the spectra nor light-curve analysis indicates a third-light contribution larger than 2%-3%. In one spectrum of OGLE LMC-ECL-12669, we noted a peculiar dimming of one of the components by 65% well outside of the eclipses. We interpret this observation as arising from an extremely rare occultation event, as a foreground Galactic object covers only one component of an extragalactic eclipsing binary.