Non-Euclidean Geometry Lesson 1b with recording.pptx -...

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Non-Euclidean Geometry Dr. Connie S. Schrock MA 722 Sections 1.5 – 1.6 Consequences and Substitutions for the Fifth
Euclid’s Fifth Postulate Postulate 5 That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. Playfair’s Axiom Through a point not on a given line there passes not more than one parallel to the line.
High School Geometry Texts create their own set of Postulates to create the same Geometry
Properties Implied by the Fifth Postulate The sum of the angles of a triangle equal two right angles. Given ABC with angles labeled as shown. Construct the line h through A so that h ││ creating angles 4 and 5. By Prop. 29 m<5 = m<3 and m<2 = <4. Using Prop. 13 we know that <1 + <4 + <5 = two right angles. Using Axiom 1 (substitution) <1 + <2 + <3 = two right angles. This is dependent on the fifth as we used a proposition after 28 that was dependent on the fifth.

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