The article reveals the trajectory of organizing and teaching a facultative course in order to familiarize students with non-Euclidean geometries in schools. It shows why students should be introduced to non-Euclidean geometries in schools, and what goals can be achieved by teaching non-Euclidean geometries. The most important aspect is the trajectory of the organization of the optional course. When organizing a facultative course, it is indicated what topics to choose, how many hours to allocate, and the program of the facultative course is developed.

Distance, metric, coordinate system, Euclidean geometry, non-Euclidean geometry

Gaybullayev N. Elements of non-Euclidean geometries at school. - Tashkent: “Teacher”, 1971.

Artikbayev A., Khatamov I. Nine geometries in the plane. - Tashkent.: “Smart Reader”. 2021.

Adajonov N.D., Yunusmetov R., Abdullayev T. Geometry part 1-2. - Tashkent.: “Teacher”. 1996.

Artikbayev A. Modern view of ancient science. - Tashkent.: “Physics, mathematics and informatics” magazine, issue 3, 2004.

Artikbayev A., Berdiyeva O. Geometry in imagination. - Tashkent.: “Physics, mathematics and informatics” magazine, issue 3, 2011.

Khatamov I., Faizullayev Sh. Interpretation of the Lobachevsky plane on a hyperboloid. - Tashkent.: “Physics, mathematics and informatics” magazine, issue 1, 2019.

Khatamov I. Development of pupils' ideas about the elements of non-Euclidean geometry. - Tashkent.: “People’s education” magazine 2020. Number 5. Page 74.

Khachaturyan A.V. Geometry of Galileo. - Moscow.: Library “Mathematical Education”, 2005.

Yaglom I.M. Galileo’s principle of relativity and non-Euclidean geometry. - Moscow: “Science”, 1969..

Atanasyan L.S. Geometry of Lobachevsky. - Moscow.: Enlightenment, 2001. 335 p.

Gaybullaev N. Formation of geometric representations of secondary school students in the study of Euclidean and non-Euclidean geometry. – Tashkent, 1972. 39 p.