Different Geometries and Parallel Postulates

Explore the realms of Euclidean, Hyperbolic, and Elliptic geometries along with their unique characteristics, axioms, and the implications of the parallel postulates. Delve into the distinctions between these geometries and the intriguing concept of mixing Euclidean and Hyperbolic geometries within the same space.

• Geometry
• Parallel Postulates
• Euclidean
• Hyperbolic
• Axioms

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1. doesnt take a stand on the PARALLEL POSTULATES

2. Ex Spherical Geometry 6 Axioms + Euclidean PP 6 Axioms + Hyperbolic PP 6 Axioms + Elliptic PP Fails to satisfy: 2. Incidence 3. Ruler Euclidean Geometry Hyperbolic Geometry Inconsistent w/ Neutral Geometry Elliptic Geometries are NOT Neutral Geometry

3. 6 Axioms + Euclidean PP 6 Axioms + Hyperbolic PP Is it possible to mix Euclidean and Hyperbolic Geometries? i.e., Within the same geometry can you have some lines and external points that have a unique parallel line and for other lines and external points with multiple parallel lines? Euclidean Geometry Hyperbolic Geometry No!

4. every every at least two lines Proof (later, maybe) negation Neutral Geometry

5. Euclidean Parallel Postulate can be proved as theorems can be proved as a theorem Euclid s Fifth Postulate

6. t l A A l (<A) + (<A ) < 180 If l and l are two lines cut by a transversal t in such a way that the sum of the measures of the two interior angles on one side of t is less than 180, then l and l intersect on that side of t.