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Intro to Euclidean Geom - Introduction to Euclidean...

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Introduction to Euclidean Geometry (Section 4.2) We now include the Euclidean Parallel Postulate as an axiom along with SMSG Postulates 1 – 15: SMSG Postulate 16: (The Euclidean Parallel Postulate) Through a given external point, there is at most one line parallel to a given line. By combining the (NIB) "Common Perpendicular" Theorem with the Alternate Interior Angles Theorem in Neutral in Neutral Geometry, there is always at least one such parallel line, so when the SMSG Postulate 16 asserts that there is at most one such parallel line, it is saying that there is exactly one such parallel line, which is what the Euclidean Parallel Postulate asserts. Thus, in Neutral Geometry, the Euclidean Parallel Postulate is equivalent to SMSG Postulate 16. Theorem 4.2.1 , The Converse of the Alternate Interior Angle Theorem : In Euclidean Geometry, if two parallel lines are crossed by a transversal, then the alternate interior angles are congruent, that is, equal in measure. (By the "EPP and AIA Converse Equivalence" Theorem, Theorem 3.4.6, since the EPP is true in Euclidean Geometry, the Converse of the Alternate Interior Angle Theorem is also true in Euclidean Geometry.) Theorem 4.2.2, The "Angle-Sums Equal 180 Degrees" Theorem : The sum of the measures of the angles in a triangle is 180º. Corollary 4.2,3, The "Euclidean Exterior Angle" Theorem : The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.
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2 Definition : A parallelogram is a quadrilateral such that the two pairs of opposite sides lie along parallel lines. Theorem 4.2.4, The "Opposite Sides of a Parallelogram" Theorem : The opposite sides of a parallelogram are congruent.
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