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

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

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