A prism whose base is a 9-sided polygon is intersected by a plane. In which case could the cross section be a 10-sided polygon?

The cross-section will be a 10-sided polygon only when the plane makes a narrow angle with the base, i.e. it will cut one base and 8 edges of the prism, resulting in there being 10 points of intersection, leaving a 10-sided polygon as the cross-section.

$12. that is my answer.