Approach: Treat the interior and exterior angles as two unknowns with a known sum and a known difference, solve both at once, then apply the diagonal formula.
Step 1: At each vertex interior $+$ exterior $=180^\circ$, and we are told interior $-$ exterior $=120^\circ$.
Step 2: Adding the two equations: $2\times\text{interior}=300^\circ\Rightarrow$ interior $=150^\circ$; subtracting gives exterior $=30^\circ$.
Step 3: Number of sides $=\dfrac{360^\circ}{\text{exterior}}=\dfrac{360}{30}=12$.
Step 4: Diagonals of an $n$-gon $=\dfrac{n(n-3)}{2}=\dfrac{12\times 9}{2}=54$.