In the half-plane model of hyperbolic geometry, we consider points on one side of a horizontal boundary line. A "line" through two points in this model is a semicircle whose center is on the boundary line. A segment is an arc on that semicircle. If one point is directly above the other, the semicircle is replaced by a vertical ray with its endpoint on the boundary line.
The figure below shows a line CD and another line through point E, parallel to CD (i.e., the lines don't intersect). How many lines do you think there are through point E parallel to CD?
This Half-Plane Model of Hyperbolic Geometry sketch depicts the hyperbolic plane and contains Custom Tools to create constructions in the upper half-plane. The Hyperbolic Triangles sketch depicts the same hyperbolic plane and contains Custom Tools for creating various centers of triangles constructed in the half-plane.
An earlier half-plane model, mentioned in Thomas Q. Sibley's Instructor's Resource Guide for The Geometric Viewpoint, is available for both Sketchpad version 4 (sketch) and Sketchpad version 3 (script files for creating lines, segments, angle measurements, and distance measurements).