abstract: For each Fuchsian triangle group of signature $(3, n, \infty), n>3$ and each $\alpha \in 0,1$ we associate a continued fraction map. For each, we construct a planar extension. We show how such extensions can be used to prove eventual expansiveness and ergodicity of the maps (with respect to the marginal measures of the planar systems). In particular, the extensions are the natural extensions of the maps. This is joint work with C. Kraaikamp, and with K. Calta.