Does a maximal planar graph with number of vertices $\ge$ 6 always have a vertex of degree exactly 5

Question

In a proof of a theorem, my professor wrote that since $G$ is a maximal planar graph with $|V|\ge6$, so, there exists a vertex of degree 5. I know the result that every planar graph has a vertex of degree atmost 5, but I am not sure about the result used by my professor. I googled it but couldn't find any such result.
Is the result true and if it is true then how to prove it ?

Answer 1

It should be incorrect. We can only say that the minimum degree is less than or equal to 5, and the maximum degree is at least 5 (using the average degree as caduk mentioned). However, we can still find counterexamples that do not contain vertices with degree 5.