In NX8.5, edit the sketch, then ( main menu) Edit - Curve - Divide - by bounding objects, pick the arc, pick the line as bounding object , indicate an approximate intersection point, then either Apply or select next bounding line + approximate intersection point.
But , since you asked me to explain, i now tried to divide / select the projected lines, and, no.
I guess that since the curves are sketch objects it's slightly more difficult to divide things than outside the sketcher since they should keep their associativity and then by the divide operation suddenly become multiple objects with associativity and potential constraints etc.
Regards,
Tomas