#### Answer

$30^{\circ}$

#### Work Step by Step

We can find the number of vertices if a regular polygon has 54 diagonals:
$\frac{n(n-3)}{2} = 54$
$n(n-3) = 108$
$n^2-3n- 108 = 0$
$(n-12)(n+9) = 0$
$n = 12~~$ or $~~n=-9$
The polygon has 12 vertices.
We can find the central angle of this polygon:
$\frac{360^{\circ}}{n} = \frac{360^{\circ}}{12} = 30^{\circ}$