Line AB touches the circle of the base of the cylinder at point A. Prove that line AB is perpendicular to the plane AOO₁A₁.

From point O, the center of the circle, draw the radius OA to the point of tangency between the circle and the straight line AB. By the punitive property, the radius OA is perpendicular to the tangent AB.
The tangent AB is perpendicular to the generatrix AA1.
Generator AA1 and radius AO have a common point A, therefore, intersecting straight lines.
If two intersecting straight lines belonging to the plane are perpendicular to the straight line, then the straight line is perpendicular to the plane. Then AB is perpendicular to AOO1O1, as required.