jmc

The sum of the interior angles in any $n$-sided polygon is $180(n-2)$ degrees, so the angle measures in a polygon with 7 sides sum to $180(7-2)=900$ degrees, which means that the desired polygon has more than 7 sides. Meanwhile, the angle measures in a polygon with 8 sides sum to $180(8-2)=1080$ degrees. So, it's possible that the polygon has $\boxed{8}$ sides, and that the last angle measures $10^{\circ }$.

To see that this is the only possibility, note that the angle measures in a polygon with 9 sides sum to $180(9-2)=1260$ degrees. Therefore, if the polygon has more than 8 sides, then the last interior angle must measure at least $1260^{\circ }-1070^{\circ }=190^{\circ }$. But this is impossible because each interior angle of a convex polygon has measure less than $180^{\circ }$.