Geometry

Geometry is the branch of mathematics concerned with questions of shape, size, the relative position of figures, and the properties of space. Built upon fundamental concepts such as points, lines, planes, angles, and solids, it uses logical deduction and rigorous proof to explore the relationships and attributes of these elements. From measuring the area of a field to modeling the curvature of the universe, geometry provides a foundational framework for understanding and quantifying the spatial world around us.

  1. Foundations of Euclidean Geometry
    1. Historical Development of Geometry
      1. Ancient Origins
        1. Euclid's Elements
          1. Modern Developments
          2. Undefined Terms
            1. Point
              1. Representation of a Point
                1. Naming Points
                  1. Point Notation
                  2. Line
                    1. Representation of a Line
                      1. Naming Lines
                        1. Line Notation
                          1. Properties of Lines
                          2. Plane
                            1. Representation of a Plane
                              1. Naming Planes
                                1. Properties of Planes
                              2. Defined Terms
                                1. Collinear Points
                                  1. Coplanar Points
                                    1. Line Segment
                                      1. Definition and Properties
                                        1. Naming Segments
                                          1. Measuring Segments
                                            1. Congruent Segments
                                            2. Ray
                                              1. Definition and Properties
                                                1. Naming Rays
                                                  1. Opposite Rays
                                                  2. Intersection
                                                    1. Intersection of Lines
                                                      1. Intersection of Planes
                                                        1. Intersection of Line and Plane
                                                      2. Postulates and Axioms
                                                        1. Euclid's Five Postulates
                                                          1. Ruler Postulate
                                                            1. Segment Addition Postulate
                                                              1. Protractor Postulate
                                                                1. Angle Addition Postulate
                                                                  1. Plane Postulate
                                                                    1. Line Postulate
                                                                      1. Parallel Postulate
                                                                      2. Theorems and Proofs
                                                                        1. Definition of a Theorem
                                                                          1. Structure of a Theorem
                                                                            1. Direct Proofs
                                                                              1. Indirect Proofs
                                                                                1. Proof by Contradiction
                                                                                2. Notation and Symbols in Geometry
                                                                                  1. Basic Geometric Symbols
                                                                                    1. Congruence and Similarity Symbols
                                                                                      1. Angle and Arc Notation

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                                                                                    2. Angles