Algumas particularidades do plano hiperbólico.

Abstract

In the early years of mathematical training, students know certain facts and take them as unshakable truths. Gradually, some of these paradigms are broken, for example with the knowledge of structures such as complex numbers, where there is a number whose square is -1. With this, the students have contact with the flexibility of Mathematics, in what relates to the possibility of constructing new sets, usually extensions of the previous sets. This, however, does not reach Geometry. The patterns of formulas and formulas that are taught remain rigid in high school and even higher education, and even for a regular undergraduate Mathematics student, the information that parallel lines determine in a common transverse congruent alternating angles is considered as immutable. The purpose of this work is to present a non-Euclidean geometry developed throughout the 19th century and has as a target the teachers of Mathematics, to show them that, just as the order of factors can change the product, not always the sum of the angles of a triangle is equal to 180 degrees.