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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 SameSide 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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 Spring '10
 Shirley
 Geometry, Angles

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