Recently, the problem of the approximate degree reduction for triangular surface attracts much attention,and is always a hotspot in the field of CAGD. This paper investigates the approximate multi-degree reduction of triangular Bézier surface by minimizing the defined distance function with GC\+1 constraint on boundary, which includes the following: 1) A kind of algorithm for the degree reduction of triangular Bézier surface is given by minimizing the defined distance function; 2) The approximate degree reduction problem of two triangular Bézier surfaces with GC\+1 constraint is studied, an algorithm of the degree reduction of two surfaces with GC\+1 constraint is proposed by adjusting the second line control vertices nearby the boundary and; 3) The approximate degree reduction of multi-triangular Bézier surface with GC\+1 constraint is studied by adjusting the internal control points. Firstly, we confirm some groups of internal control points after each two triangular Bézier surfaces approximate degree reduction with GC\+1 constraint, and then structure blending function and constructing a new blending format for approximating multi-degree reduction surface. Finally, It is proved in theory that the new triangular Bézier surface and its surrounding surfaces still keep GC\+1 constraint. The simulation results prove that the proposed algorithm is practical and efficient.