The boron and oxygen core level RIXS (Resonant Inelastic X-ray Scattering) spectra of LiB3O5 (LBO) and the β phase of BaB2O4 (BBO) have been measured. The RIXS data confirm several band structure calculations and verify the existence of two different oxygen environments in BBO. A boron core exciton level exists in both LBO and BBO. The RIXS spectra excited in the vicinity of the B 1s core resonance show two principal features: the scattering on a valence excitation and scattering on a core excitation. An energy loss sideband to the elastic scattering peak is present when the core exciton is created. The energy loss shoulder appears to result from lattice relaxation in the absorption site.