Each compact connected $2$-manifold $S$ has a proper triangulation $K$, so we can order all $2$-simplices of $S$, $F_1,F_2,\ldots,F_{k-2}$ such that $F_i$ meets $F_{i-1}\cup F_{i-2} \cup \ldots \cup F_1$ in at least one $1$-simplex in $K$ for all $i>1$.
My question arises from the next step of the proof, which I feel somewhat not rigorous. The book say that we can first choose $F_1$ and map(by some isomorphic simplicial map) it into the plane. Next, map (by some isomorphic simplicial map) $F_2$ into the plane so that the image intersects that of $F_1$ in a $1$-simplex, which is the image of the common edge of $F_1$ and $F_2$ under the former simplicial map. Continuing in this way, we can yield a polygon in the plane consisting $k-2$ $2$-simplices, say $G_1,G_2,\ldots,G_{k-2}$, obtained from this sequence of isomorphic simplicial maps in such a way that $G_i$ meets $G_{i-1}\cup G_{i-2}\cup\ldots\cup G_1$ in precisely in $1$-simplex for all $i>2$ and hence the polygon so obtained has $k$ edges.
But I think we should add the followings: In each of these steps, when we map a $2$-simplex by some isomorphic simplicial map in the triangulation sapce into the plane, we have to make some alternative in the edge of the image (maybe twist it but remains linear) so that it would not overlap the the other $2$-simplices in the plane or intersect them in two $2$-simplices. Then the result that the polygon so obtained has $k$ edges will then be true. I feel my argument is somewhat naive and not rigorous. How should I improve it?