We theoretically calculate the composition dependence of the valence-and conduction-band discontinuities at the interfaces between selected III-nitride ternary materials with wurtzite structure, e.g. AlxGa1-xN/AlxGa1-xN, InxGa1-xN/InxGa1-xN and InxA1-xN/InxA1-xN. Calculations are performed using a theoretical model, initially proposed by S. L. Chuang et al. [1]. Depending on a particular set of input parameters, simulation results show that band offsets change more or less with strain. The valence band offsets, together with the resulting conduction band offsets, indicate that a type-I, type-II band line-up forms at InxGa1-xN/GaN, AlxGa1-xN/GaN heterojunctions with varying In, Al contents respectively. Also, based upon the same model, we propose a type I Indium-dependent band alignment in InxAl1-xN/AlN interfaces. The failure of the transitivity rule, which is often used to determine the band offsets in heterojunctions, was demonstrated and its cause was explained. The obtained results are well compared with experiment and theory in various reliable test cases and therefore provide a basis for optimization and design of novel interface structures.