In other words, consistency of Euclidean geometry implies consistency of the geometry of Lobachevsky. The construction is now known as Beltrami-Klein model of Lobachevsky geometry. Sometimes it's called the projective model because it may be extended to conic sections and builds on an unusual usage of usual straight lines. The Beltrami-Klein model considers the region strictly inside a circle as a model of a plane. Note that the region does not include points on the circle itself.
Lines are chords connecting points on the circle with the endpoints excluded.
It's very easy to verify that the first four Euclid's postulates hold but there is infinitely many lines through a given point and parallel to a given line. There might be an uneasy feeling when calling a segment a line and a disk a plane. Poincare suggested a dictionary method. In a dictionary, most of the entries have several interpretations. Imagine a being that uses the same words as we do but imputes to them a different meaning.