Section 7.1: Geometry

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7.1 Outline

  1. Greek (Euclidean) geometry
    1. undefined terms
      1. point
      2. line
      3. plane
      4. surface
    2. categories
      1. traditional (Euclidean) geometry
      2. transformational geometry
    3. Euclid’s postulates
      1. postulate
      2. axiom
      3. theorem
      4. five postulates
    4. parallel lines
    5. non-Euclidean geometries
    6. straightedge
    7. line segment
    8. congruent figures
    9. construct a figure
      1. construct a circle
      2. construct a line parallel to a given line through a given point
  2. Transformational geometry
    1. transformation
    2. reflection
    3. line of symmetry
  3. Similarity
    1. definition
    2. similar

 

7.1 Essential Ideas

Geometry can be separated into two categories:

  1. Traditional (which is the geometry of Euclid)
  2. Transformational (which is more algebraic than the traditional approach)

When Euclid was formalizing traditional geometry, he based it on the following five postulates:

  1. A straight line can be drawn from any point to any other point.
  2. A straight line extends infinitely far in either direction.
  3. A circle can be described with any point as center and with a radius equal to any finite straight line drawn from the center.
  4. All right angles are equal to each other.
  5. Given a straight line and any point not on this line, there is one and only one line through that point that is parallel to the given line.