Almost 170

It's not so difficult to find a solution for $35$ .

Suppose $36$ is still possible. If the polygon has $k$ number of sides, then the sum of its angles is $(k-2)\times 180°$ . From that $(k-2)\times 180° < 36\times 170° + (k-36)\times 180° \\ \Rightarrow 34\times 180°<36\times 170°$ which is definitely false.

Therefore the answer is $\boxed{35}$ .