The Triangle Splitting iteration method is an effective iteration method for solving large-scale sparse non-Hermitian positive definite system of linear algebraic equations. By making use of the Triangle Splitting iteration method on non-Hermitian positive definite matrices as the inner solver of the inexact Newton method, we establish a class of inexact Newton-Triangle Splitting iteration methods for solving the large-scale sparse system of nonlinear algebraic equa-tions with positive definite Jacobian matrices in the paper. For this class of inexact Newton methods, two types of local convergence theorems are proved under proper conditions. The numerical results are given to examine their feasibility and effectiveness. The numerical implementations also show that the Newton-Triangle Splitting methods have advantages over Newton-BTSS methods with less computation time and iteration steps.