Example of a equiangular and not regular quadrilateral in the hyperbolic plane

Question

I would like to see a quadrilateral in the hyperbolic plane which has $4$ equal angles but is not regular. Can someone tell me an example?

I know that this is impossible for triangles, but I think that this should be different for quadrilaterals. Unfortunately I could not find any book where this is explained.

Best wishes

Answer 1

Take a line segment $PQ$. Draw segments perpendicular to the endpoints, with equal distances $\delta$ above $PQ$ and below $PQ$, making a letter $H.$ Join the pairs of $H$ endpoints, resulting in a sort of number $8$ or capital $\theta$. This quadrilateral has four equal angles. Unless $\delta $ is a very special value, this is not regular.