I have two questions: What is the special case of parallels when the Klein model will not create a perpendicular?
II) From point P, drop a perpendicular Q with line l. Consider the hyperbolic angle <QST. T is a point on the hyperbolic ray PB.
Use the image to explain why <QST (the angle of parallelism) cannot be a right angle. Then use this result, and the fact that AD and BC are limiting parallels, to show that the angle of parallelism cannot be greater than a right angle.
Image of Klein Model 