We define two types of the α-Farey maps Fα and for, which were previously defined only for by Natsui (2004). Then, for each, we construct the natural extension maps on the plane and show that the natural extension of is metrically isomorphic to the natural extension of the original Farey map. As an application, we show that the set of normal numbers associated with α-continued fractions does not vary by the choice of α,. This extends the result by Kraaikamp and Nakada (2000).