Plane Geometry - Bruce E. Shapiro
Plane Geometry - Bruce E. Shapiro Plane Geometry - Bruce E. Shapiro
136 SECTION 29. TRANSVERSALSCorollary 29.9 (Existence of Parallel Line) If l is a line and P ∉ l isa point, then there is a line m ‖ l with P ∈ m.Proof. Drop a perpendicular from P to l (existence of perpendiculars, theorem21.5).Call the foot of the perpendicular Q and define t = ←→ P Q.By the protractor postulate (axiom 16.2), we can construct a line m ⊥ tthrough P .By the interior angle theorem, m ‖ l.Recall that the elliptic parallel postulate says that no parallel lines exist.Since we have just proven that parallel lines exist in neutral geometry, weknow that this postulate is false.Corollary 29.10 The elliptic parallel postulate is false in neutral geometry.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
Section 30Triangles in NeutralGeometryDefinition 30.1 The angle sum of △ABC is defined asσ(△ABC) =µ(∠CAB) + µ(∠ABC) + µ(∠BCA)Figure 30.1: Theorem 30.2 says that β + γ < 180.Theorem 30.2 The sum of the measure of any two angles in a triangle isless than 180: For any triangle △ABC,µ(∠CAB) + µ(∠ABC) < 180Proof. Let D ∈ ←→ AB such that A ∗ B ∗ D (see figure 30.1).By the linear pair theorem (thm. 18.4), α + β = 180.137
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136 SECTION 29. TRANSVERSALSCorollary 29.9 (Existence of Parallel Line) If l is a line and P ∉ l isa point, then there is a line m ‖ l with P ∈ m.Proof. Drop a perpendicular from P to l (existence of perpendiculars, theorem21.5).Call the foot of the perpendicular Q and define t = ←→ P Q.By the protractor postulate (axiom 16.2), we can construct a line m ⊥ tthrough P .By the interior angle theorem, m ‖ l.Recall that the elliptic parallel postulate says that no parallel lines exist.Since we have just proven that parallel lines exist in neutral geometry, weknow that this postulate is false.Corollary 29.10 The elliptic parallel postulate is false in neutral geometry.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
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