New moderate-resolution mid-infrared spectroscopy at 10 mu m of 27 infrared galaxies is presented. The galaxies have been chosen from three 60-mu m selected and one 12-mu m selected complete flux-limited catalogues of galaxies; 17 of these sources have L(IR)(8-1000 mu m) greater than or equal to 5 x 10(11) L(.). A high-resolution spectrum of the source Arp 299B1 is also presented. Combining these new results with previously published results, a nearly complete 60-mu m selected flux-limited subsample, with L(IR)(8-1000 mu m) greater than or equal to 1.6 x 10(11) L(.), of 25 galaxies is defined, Within this subsample, it is found that the dominant power sourer of infrared galaxies in the luminosity range 1.6 x 10(11) < L(IR)(8-1000 mu m) < 10(12) L(.) is massive star formation based on the clear presence of the 11.3-mu m aromatic hydrocarbon emission feature in the majority of the spectra. Three of the five ultraluminous infrared galaxies [L(IR)(8-1000 mu m) greater than or equal to 10(12) L(.)] within this subsample show evidence that an active galactic nucleus provides an energetically important power source based on the detection of silicate absorption in their mid-infrared spectra, The physical basis of a possible anticorrelation between the 11.3-mu m feature equivalent width and infrared light to molecular gas mass ratios is discussed.