Quiz

A cycle isn't necessary. It's not required to end where you started!

And vertex C also has an odd degree!
That's true, but there should still be a situation where a single edge would create an Eulerian cycle, but since we have 3 vertices with odd degrees, even if we connect two of them with an edge, one will still have an odd degree. This proves that it's impossible!!!

Added 4 minutes later
That is, even if we start in rectangle B and end in rectangle D (or vice versa) (connect them with an edge), since 5 edges lead to vertex C, we will still have to traverse one of them twice or not at all. This proves the theorem about the Eulerian cycle.