Transversals in plane hyperbolic geometry

In a plane hyperbolic geometry (in the sense of F. Bachmann's book ' Aufbau der Geometrie aus dem Spiegelungsbegriff') we prove that an asymptotic n-gon does not admit a transversal when n is odd; the assumption of n being odd is crucial. For each n one finds an asymptotic n-gon that does not have a transversal.