This paper proposes and applies a method to sort two-dimensional control points of triangular Bézier surfaces in a row vector. Using the property of bivariate Jacobi basisfunctions, it further presents two algorithms for multi-degree reduction of triangular Bézier surfaces with constraints, providing explicit degree-reduced surfaces. The first algorithm can obtain the explicit representation of the optimal degree-reduced surfaces and the approximating error in both boundary curve constraints and corner constraints. But it has to solve the inversion of a matrix whose degree is related with the original surface. The second algorithm entails no matrix inversion to bring about computational instability, gives stable degree-reduced surfaces quickly, and presents the error bound. In the end, the paper proves the efficiency of the two algorithms through examples and error analysis.