Accepted manuscript ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA https://doi.org/10.26493/1855-3974.2740.7ab (Also available at http://amc-journal.eu) S 2 coverings by isosceles and scalene triangles – adjacency case II * Catarina P. Avelino † Center of Mathematics of the University of Minho – UTAD Pole(CMAT-UTAD), University of Tr´ as-os-Montes e Alto Douro, Vila Real, Portugal affiliated also with: Center for Computational and Stochastic Mathematics (CEMAT), University of Lisbon (IST-UL), Portugal Altino F. Santos Center of Mathematics of the University of Minho – UTAD Pole(CMAT-UTAD), University of Tr´ as-os-Montes e Alto Douro, Vila Real, Portugal Received 4 April 2019, accepted 6 January 2022 Abstract The aim of this paper is to complete the study and classification of spherical f-tilings by scalene triangles T and isosceles triangles T ′ within a subclass defined by the adjacency of the lower side of T and the longest side of T ′ . It consists of eight families of f-tilings (two families with one continuous parameter, one family with one discrete parameter and one continuous parameter, and five families with one discrete parameter). We also analyze the combinatorial structure of all these families of f-tilings, as well as the group of symmetries of each tiling; the transitivity classes of isogonality are included. Keywords: Dihedral f-tilings, combinatorial properties, spherical trigonometry. Math. Subj. Class. (2020): 52C20, 52B05, 20B35 1 Introduction A folding tessellation or folding tiling (f-tiling, for short) of the sphere S 2 is an edge-to- edge finite polygonal tiling τ of S 2 such that all vertices of τ satisfy the angle-folding * This research was partially financed by Portuguese Funds through FCT (Fundac ¸˜ ao para a Ciˆ encia e a Tec- nologia) within the projects UIDB/00013/2020 and UIDP/00013/2020 of CMAT-UTAD, Center of Mathematics of University of Minho, and projects UIDB/04621/2020 and UIDP/04621/2020 of CEMAT/IST-ID, Center for Computational and Stochastic Mathematics, Instituto Superior T´ ecnico, University of Lisbon. † Corresponding author. E-mail addresses: cavelino@utad.pt (Catarina P. Avelino), afolgado@utad.pt (Altino F. Santos) cb This work is licensed under https://creativecommons.org/licenses/by/4.0/