In the approach taken here, two geometric figures are defined to be congruent if there is a sequence of rigid motions that carries one onto the other. Reflections and rotations each explain a particular type of symmetry, and the symmetries of an object offer insight into its attributes-as when the reflective symmetry of an isosceles triangle assures that its base angles are congruent. Fundamental are the rigid motions: translations, rotations, reflections, and combinations of these, all of which are here assumed to preserve distance and angles (and therefore shapes generally).
The concepts of congruence, similarity, and symmetry can be understood from the perspective of geometric transformation. Later in college some students develop Euclidean and other geometries carefully from a small set of axioms. (Spherical geometry, in contrast, has no parallel lines.)ĭuring high school, students begin to formalize their geometry experiences from elementary and middle school, using more precise definitions and developing careful proofs. Euclidean geometry is characterized most importantly by the Parallel Postulate, that through a point not on a given line there is exactly one parallel line. An understanding of the attributes and relationships of geometric objects can be applied in diverse contexts-interpreting a schematic drawing, estimating the amount of wood needed to frame a sloping roof, rendering computer graphics, or designing a sewing pattern for the most efficient use of material.Īlthough there are many types of geometry, school mathematics is devoted primarily to plane Euclidean geometry, studied both synthetically (without coordinates) and analytically (with coordinates).
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