In this work, we give a complete classification of spherical dihedral f-tilings when the prototiles are two noncongruent isosceles triangles with certain adjacency pattern. As it will be shown, this class is composed by two discrete families denoted by {ieɛ}, m ≥ 2, m ∈ ℕ, F, k ≥ 4, k ∈ ℕ and two sporadic tilings denoted by G and H.