Maximal consistent sets are complete

I'm stuck on a problem on the theory of structures and essentially boiled it down to this: If $\displaystyle \Gamma$ is a maximal consistent set of sentences ($\displaystyle \forall \phi \in \text{Sent}(\mathcal{L}), \Gamma \vdash \phi$ or $\displaystyle \Gamma \vdash \neg \phi$, but not both) is complete ($\displaystyle \forall \phi \in \text{Sent}(\mathcal{L}), \phi \in \Gamma$ or $\displaystyle \neg \phi \in \Gamma$). I've seen another definition for maximal consistency which says that there are no proper consistent extensions of $\displaystyle \Gamma$ and that this is equivalent to being complete, but I fail to see an equivalence between these two definitions. (Thinking)

Please help?