The study of dihedral f-tilings of the Euclidean sphere _S_2 by triangles and r-sided regular polygons was initiated in 2004 where the case r = 4 was considered . In a subsequent paper , the study of all spherical f-tilings by triangles and r-sided regular polygons, for any r ≥ 5, was described. Later on, in  the classfication of all f-tilings of _S_2 whose prototiles are an equilateral triangle (a 3-sided regular polygon) and an isosceles triangle are obtained. In this paper we extend these results presenting the study of all spherical f-tilings by equilateral triangles and scalene triangles in a particular way of adjacency. For this particular case of adjacency, we get two continuous families of f-tilings denoted by _F_1δ and _F_2δ. The combinatorial structure including the transitive properties under the action of their symmetry groups is given in Table 1.
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Ana M. d'Azevedo Breda, Patrícia S. Ribeiro, Altino F. Santos, Dihedral F-Tilings of the Sphere by Equilateral and Scalene Triangles-I. Rend. Sem. Mat. Univ. Padova 124 (2010), pp. 1–23DOI 10.4171/RSMUP/124-1