Formation of the concept of an angle in the school course of mathematics by means of metric geometry

Abstract

In the school mathematics course, the corner is systematically studied from the fifth grade. Surprisingly, this concept still has no fixed definition, or at least no description. Even in higher mathematics, it is interpreted differently, not to mention school textbooks. The angle is perceived both as a line and as part of a plane, and sometimes the angle is identified with its numerical characteristic. This perception of the angle, in our opinion, is due to its widespread use in various fields of science and technology.
This paper proposes the definition of an angle as an ordered triple of points. This definition, in our opinion, is a logical addition to the existing definitions of angle. In addition, this approach to the concept of angle makes it possible to use elements of metric geometry to study its properties, as well as to use the angle to define the concept of rectilinear placement of points. The application of the basic concepts of metric geometry to the study of the properties of angle makes it possible to acquaint pupils with the elements of non-Euclidean geometry, which is currently completely absent in school textbooks on geometry, including textbooks for classes with depth study of mathematics.
The introduction of the generalized concept of angle in the school course of geometry, in our opinion, should begin with a demonstration of examples that indicate the relativity of basic geometric concepts - point, straightness, distance, angle. Such examples will prepare students for the adequate perception in the future of the basic concepts and relations of non-Euclidean geometries. The use of metric geometry for this eliminates the need to consider a significant number of axioms, because it uses only three axioms of distance, which are intuitive to students.
The paper presents a number of examples that can be considered in lessons in classes with advanced study of mathematics, as well as in extracurricular activities in mathematics classes, or in electives in mathematics.