More on Neutral Geometry II - More on Neutral Geometry II...

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More on Neutral Geometry II (Including Sections 3.4 and 3.5) Section 3.4: The Place of Parallels Theorem 3.4.1 – The "Alternate Interior Angles (AIA)" Theorem or also the "Congruent Alternate Interior Angles Make Parallel Lines" Theorem: Two lines may or may not be parallel at first glance, but if at least one transversal of the lines has congruent alternate interior angles, then the lines are indeed parallel. (Pay attention to the way that the proof uses the Exterior Angle Theorem.) Note: The Converse of the Alternate Interior Angle Theorem is NOT TRUE in Hyperbolic Geometry, so the Converse cannot be proven in Neutral Geometry! Corollary 3.4.2 - The "Common Perpendicular makes Parallel Lines" Theorem: Two lines which are perpendicular to the same line are parallel lines. Similar to the "Alternate Interior Angles" Theorem (Thm 3.4.1) are the following: Corollary 3.4.3 - The "Congruent Corresponding Angles" Theorem. If two lines have a transversal such that a pair of corresponding angles formed are congruent, then the two lines must be parallel. Corollary 3.4.4 - The "Supplementary Same-Side Interior Angles" Theorem: If two lines have a transversal such that a pair of interior angles on the same side of the transversal are supplementary, then the two lines must be parallel. DEFINITION: The Euclidean Parallel Postulate (EPP) : For every line l and every point P not on line l , there is one and only one line m that contains P and is parallel to l . (This formulation of the EPP is sometimes called “Playfair’s Postulate.”)
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More on Neutral Geometry II - More on Neutral Geometry II...

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