A decade long experience of teaching the course Fundamental of Geometry at the University of North Carolina at Charlotte, many notes for exercises, many hours of extra reading and careful research are the bases for this bulky work. The present fifth volume recalls Hilbert's axioms from the Foundations of Geometry, and elaborates hyperbolic geometry. Here the disk models of Poincaré and Klein are used to do a lot of constructions, using straightedge and compass from the background Euclidean geometry. Too, Hilbert's axiomatic approach based on the asymptotic rays, is explained from the beginning up to the reconstruction of the Poincaré disk model. The last section gives a short course on Gauss' differential geometry and the pseudo sphere.