Since a disk doesn't contain a Möbius or any other non orientable surface, it is orientable. I want to prove it rigorously by showing every triangulation on a disk is orientable. For this, I was trying to use an induction. For example, If $D=T_1\cup T_2 \cdots\cup T_n$ is a triangulation on a disk, then we can construct $D$ by edge pairing the triangles. Thus, from the facts $T_1$ is orientable and $T_1\cup T_2 \cdots\cup T_j$ is orientable, I one to deduce that the whole triangulation is orientable. However, since the ways of edge pairing are not unique, I cannot make progress.
Any help will be appreciated.