Planar graphs and bipartite graphs

Question

So, I'm supposed to prove by induction on the number of edges that a planar graph is bipartite if and only if every face has even length. I can only think on adding new vertices to $G$ to prove that those conditions will still be true as step (as I'm assuming the base would be something like the graph $G = (2,1)$). Because by incrementing the number of edges, I feel that it is easy to the graph have a subset that is equals to $K_{3,3}$, which will make the statement invalid. Thank you.

Answer 1

"Planar" is a supposition of this theorem, not something you need to prove. If you add an edge that makes the graph non-planar, it does not contradict the theorem. It just means you did something pointless, since you've arrived at a graph to which the theorem does not apply.

The inductive step here is to assume the theorem is true for all for all planar graphs with fewer than $n$ edges, and you have a planar graph with $n+1$ edges. Remove an edge from this graph and show from the bipartiteness of the reduced graph whether this graph is bipartite or not.