Prove that the graph dual to Eulerian planar graph is bipartite. How would I go about doing this proof I am not very knowledgeable about graph theory I know the definitions of planar and bipartite and dual but how do you make these connection

So $G$ is planar and eulerian. We must prove $G'$ is bipartite. Asume $G'$ is not bipartite. Now I want you to forget about the fact that $G'$ is the dual of $G$. Just think of $G'$ as a normal graph in which the vertices of $G'$ are drawn as vertices and not as the faces of $G$.
Since $G'$ is not bipartite it has an odd cycle, one of the faces inside that odd cycle must therefore have an odd number of edges. That face is a vertex of odd degree in $G''$,so $G''$ is not eulerian. Now,$G\cong G''$ so $G$ is not eulerian, a contradiction. The contradiction comes from assuming $G'$ is not bipartite.
A key step is the fact $G\cong G''$.
