We report on the observation of the strong-coupling regime between quantum well excitons and a high incidence "Brewster cavity mode" previously identified as the generalized Brewster angle condition in multilayer structures [H. F. Mahlein, J. Opt. Soc. Am. 64, 647 (1974)JOSAAH0030-394110.1364/JOSA.64.000647]. This propagating mode is inside the light cone and therefore can be accessed from the top side of the sample without the need for prism or grating coupling methods. The observed anticrossing is clear evidence of the strong light-matter coupling regime. All the results are accurately reproduced by transfer matrix simulations. These results demonstrate the high potential of such structures for the study of propagating polaritons at high k, which could be harnessed for the realization of polaritonic circuit devices.