This framework deals with noneuclidean geometry, respectively one of its parts - hyperbolic geometry as geometry contrary to the euclidean one. The focus of Section 1 lies on the parallel postulate, which began the main interest of many mathematicians through the history. The non intuitiveness of the postulate leads to many unexpected results and even to the foundation of new noneuclidean geometry. Although it can be shown that this new type of geometry is equally consistent as the euclidean one, because of the strong beliefs that the euclidean one describes rules of the real world, hyperbolic geometry didn’t get the popularity which it deserves. Sphere, elliptical and hyperbolic geometries are considered to be noneuclidean geometries, but this framework deals only with hyperbolic geometry, which is the main subject of Section 2. Considering that the parallel postulate was the main reason for the foundation of the new geometry, Section 2 begins with a definition of parallel and hyperparallel lines. This different definition of line parallelism leads us to the result that the sum of angles in a triangle is less than 180º. Not only is it less, but it isn’t even the same for all triangles. The best part of this section deals with trigonometric equations. There are presented a sine and cosine formulas equivalent to the one in the euclidean geometry. From the result th α = cos α it follows \( \tan \frac{\pi (p))}{2}= e^{-\frac{p}{k}}\) which shows that for k → ∞ angle of parallelism becomes \(\frac{\pi }{2}\) , meaning that the euclidean geometry appears as a special case of hyperbolic geometry. This way hyperbolic geometry obtained a long desired use because it opens up the possibility that the rules of the real world are reflected exactly the same in hyperbolic geometry.