2.2 Fundamental axioms and postulates of Euclidean Geometry

Euclidean geometry forms the basis of our understanding of space and shapes. It's built on five fundamental postulates, with the parallel postulate being the most famous and controversial. These postulates allow us to construct and prove geometric truths.

While Euclidean geometry works well for everyday situations, it has limitations. It assumes a flat plane and doesn't account for curved spaces. This led to the development of non-Euclidean geometries, which better model certain real-world scenarios.

Foundations of Euclidean Geometry

Fundamental postulates of Euclidean Geometry

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• Postulate 1: Any two distinct points can be joined by a unique straight line segment
• Postulate 2: A straight line segment can be extended indefinitely in either direction to form a straight line
• Postulate 3: A circle can be constructed with any given point as its center and any given line segment as its radius
• Postulate 4: All right angles are congruent meaning they have equal measure
• Postulate 5 (Parallel Postulate): If a straight line intersects two other straight lines forming interior angles on one side that sum to less than two right angles, then the two straight lines, if extended indefinitely, will intersect on that side
  • Equivalent statement in a plane, given a line and a point not on it, there is at most one line parallel to the given line that passes through the point

Axioms vs postulates vs theorems

• Axioms:
  • Statements accepted as true without proof because they are self-evident
  • Equality axiom things equal to the same thing are equal to each other
• Postulates:
  • Statements assumed to be true without proof for a specific mathematical system
  • Serve as the foundation and starting point for deriving other geometric truths
• Theorems:
  • Propositions that can be logically proven using axioms, postulates, definitions, and previously proven theorems
  • Triangle angle sum theorem the sum of the angles in any triangle equals 180°

Application of geometric fundamentals

• Example 1: Constructing an equilateral triangle using Postulates 1 and 3
  1. Draw a line segment $\overline{AB}$
  2. With A as center and $\overline{AB}$ as radius, draw a circle
  3. With B as center and $\overline{AB}$ as radius, draw another circle
  4. Label the intersection points of the circles as C
  5. Connect A, B, and C to form $\triangle{ABC}$, an equilateral triangle
• Example 2: Proving base angles of an isosceles triangle are congruent using Postulates 2 and 5
  • Given $\triangle{ABC}$ with $\overline{AB} \cong \overline{AC}$
  • Extend $\overline{BA}$ to $\overline{BD}$ and $\overline{CA}$ to $\overline{CE}$ such that $\overline{BD} \cong \overline{CE}$
  • Draw $\overline{DC}$ forming congruent triangles $\triangle{ABD} \cong \triangle{ACE}$ by SSS
  • $\angle{ABC} \cong \angle{ACB}$ since they are corresponding parts of congruent triangles

Limitations and Assumptions

Limitations of Euclidean assumptions

• Assumes a perfectly flat, two-dimensional plane which does not always model real-world geometry
• The Parallel Postulate is not self-evident and has been challenged, leading to non-Euclidean geometries
  • Hyperbolic geometry allows infinitely many lines through a point parallel to a given line
  • Elliptic geometry has no parallel lines at all
• Does not account for the curvature of space, a key concept in general relativity
• Based on idealized notions of points, lines, and planes that do not truly exist in physical reality