Euclidian plane geometry, also called flat or parabolic geometry, is the theory of points, lines and angles on a plane. Basic non-Euclidian geometry includes two-dimensional spherical geometry; three-dimensional hyperbolic or Lobachevsky-Bolyai-Gauss geometry; and three-dimensional elliptic, or Reimannian geometry. Teachers still need mastery of these concepts and principles to teach plane geometry.
The axioms (rules) of Euclidian geometry have provided the foundation for the study of plane geometry since 300 B.C. In the 19th century, several mathematicians introduced the concepts of congruence, continuity and "betweenness." Teachers must be able to prove theorems about the concepts of plane geometry, understand the importance of axiomatic reasoning (logic following the rules of Euclidian geometry) and be alert for new discoveries.
Teachers of plane geometry must exhibit proficiency of the concepts involved in transformational geometry (changing the position of a shape on a plane), coordinates (description of the location of a point on a plane using ordered pairs of numbers) and vectors (a mathematical structure that represents both magnitude, or distance, and direction, used to represent such things as the wind or other moving forces).
Trigonometry studies the relationship of angles, and their relationships to plane shapes and three-dimensional figures. Skills in trigonometry are especially important for teachers of high school and college geometry.
Important concepts and skills in plane geometry includes tiling (tessellation), fractals (geometric shapes exhibiting symmetry of scale), computer graphics, robotics and visualization. Familiarity with these concepts and skills, and the ability to engage student interest by relating them to real-life applications, is an important aspect of teaching plane geometry.
Dynamic drawing tools, such as Cabri Geometry and Geometer's Sketchpad, help students investigate scientific and artistic problems in computer graphics. The knowledge they gain translates to real-life activities. Experience with these tools enhances a teacher's skills in working with coordinates and representations, enabling them to better prepare the students with interests in these areas.