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This book is mainly intended to college and undergraduate mathematics major students as well as a reference for mathematics teachers to fill the gap in the area of geometry. And as basic reference book it has the following objective shall be meet using the book as reference:Objective: At the end of this book, the reader should be able to:- understand the basic notions in absolute geometry,- apply concepts of algebraic geometry in Euclidean and hyperbolic geometry,- apply the distance function and related concepts to prove the congruence between triangles,- understand the basic axioms of Euclidean geometry and its consistency,- apply axioms and theorems to solve different problems,- understand the basic axioms and unique properties of hyperbolic geometry and its consistency,- understand The Poincare Model,- distinguish the difference between Euclidean and hyperbolic geometry,- develop skills in mathematical proofs.