The classification of the dihedral folding tessellations of the sphere and the plane whose prototiles are a kite and an equilateral triangle were obtained in a recent paper, [#!C_A_2012!#]. Concerning to isosceles triangles the classification is much harder, which is not surprising since more angles are involved. In this paper we extend this classification presenting all the dihedral folding tessellations of the sphere by kites and isosceles triangles in a particular case of adjacency. A list containing these tilings including its combinatorial structure is presentes in Table 1.

CEMAT - Center for Computational and Stochastic Mathematics