Grafteori/ Kombinatorik Faces

If I get the definitions right the frontier of a face in a graph is the edges seperating the face from the other faces. But is the frontier said to be a "subset" of the face?

I want to show that in every plan graph, for half of the faces, every edge of a cycle is in a face. I know that for every edge in a cycle, the edge will be on the frontier of 2 faces. Any advice on how to get the whole way?