#### Answer

$36^{\circ}$

#### Work Step by Step

We can find the number of vertices if a regular polygon has 35 diagonals:
$\frac{n(n-3)}{2} = 35$
$n(n-3) = 70$
$n^2-3n- 70 = 0$
$(n-10)(n+7) = 0$
$n = 10~~$ or $~~n=-7$
The polygon has 10 vertices.
We can find the central angle of this polygon:
$\frac{360^{\circ}}{n} = \frac{360^{\circ}}{10} = 36^{\circ}$