This is the face version of the Handshake Lemma.

Face Handshake Lemma

Each face $F$ has a boundary that is some loop $L$. We can define

$$\text{Sides}(F)=|E(L)|$$

by following the edges on the loop for some face. So, if we could every face,

$$\sum _F\text{Sides}(F)=2|E|$$

which is the Face Handshake Lemma.