## Elementary Geometry for College Students (6th Edition)

$30^{\circ}$
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: $\frac{360^{\circ}}{n} = \frac{360^{\circ}}{12} = 30^{\circ}$