Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-07-03T19:23:35Z No. of bitstreams: 1 josecarlosvalerio.pdf: 982623 bytes, checksum: 72d4cd36b83464bfd6a83caee289315d (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-08-08T13:23:10Z (GMT) No. of bitstreams: 1 josecarlosvalerio.pdf: 982623 bytes, checksum: 72d4cd36b83464bfd6a83caee289315d (MD5) / Made available in DSpace on 2017-08-08T13:23:10Z (GMT). No. of bitstreams: 1 josecarlosvalerio.pdf: 982623 bytes, checksum: 72d4cd36b83464bfd6a83caee289315d (MD5) Previous issue date: 2017-05-04 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Na presente dissertação será introduzido o desenvolvimento histórico da Geometria Hiperbólica. Será apresentado o quinto postulado de Euclides, de acordo com o ponto de vista dos Axiomas de Hilbert, correlacionando-os com os resultados da Geometria Neutra. Serão apresentados e provados alguns resultados da Geometria Hiperbólica, no que diz respeito às propriedades das retas paralelas, dos triângulos generalizados e seus critérios de congruência. Por fim, serão discutidas as propriedades que são válidas tanto para a Geometria Euclidiana quanto Hiperbólica, enfatizando que a principal diferença entre elas é o postulado das paralelas. / In the present dissertation we will introduce the historical development of the hyperbolic geometry. We will present Euclid’s fifth postulate from the Hilbert’s axioms point of view and we will correlate them with results of the Neutral Geometry. We will present and prove some results of the Hyperbolic Geometry, regarding the properties of the parallel lines, and the generalized triangles and their congruence criteria. At last, we will discuss the proprieties which are valid in both Euclidean and Hyperbolic Geometry, and we will emphasize that the main difference between them is the parallel postulate.