In this dissertation , after a historical review of non-Euclidean geometry and Hilbert axioms for Euclidean plane, the notions of K-loops and gyrogroups and then the gyrovector space approach of A . A . Ungar are introduced . The role of gyrovector spaces for hyperbolic geometry is the same as the role of vector spaces for Euclidean geometry . Considering the notion of {\\it quasi-end} in general (i.e . including non-continuous and non-Archimedean) absolute planes , gives us a complete justify; MARGIN: 0cm 0cm 0pt" incident with a line . A . A . Ungar introduced gyrogeometry in full analogy with Euclidean geometry based on Einstein's relativity addition. Another approach is from H . Karzel. We develop a formulary for point reflections , addition , defect and area in the Beltrami-Klein model of hyperbolic geometry with geometric approach of Karzel. Also using the isomorphism between the ordered fields and enables us to establish a uniform approach to trigonometry in Beltrami-Klein model in full analogous with Euclidean Geometry . At the end , we present some applications of hyperbolic geometry in the Einstein's theory of special relativity .