Plane Geometry - Bruce E. Shapiro

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166 SECTION 32. THE EUCLIDEAN PARALLEL POSTULATEBy Wallis’ Postulate there exists a point T such that △P RS ∼ △P QT .Since ∠P QT = ∠P RS = 90 then T ∈ l.Since ∠QP T = ∠RP S then T ∈ n.Hence n ∩ l ≠ ∅ and therefore n ̸‖ l. Thus we must reject the RAAhypothesis and conclude that n = m, i.e., the parallel line is unique. ThusWallis’ Postulate ⇒ the Euclidean Parallel Postulate.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
Section 33RectanglesDefinition 33.1 A rectangle is a quadrilateral each of whose angles areright angles.Definition 33.2 A square is a rectangle each of whose sides has equallength.Theorem 33.3 The following statements are equivalent to the Euclideanparallel postulate:1. There exists a triangle whose defect is 0.2. There exists a right triangle with defect 0.3. (Clairut’s Axiom.) There exists a rectangle.4. There exists arbitrarily large rectangles.5. The defect of every right triangle is 0.6. The defect of every triangle is 0.Lemma 33.4 Let △ABC be a triangle. Then at least two of its interiorangles are acute.Proof. Suppose that two of the interior angles are not acute. Call theseangles α and β, and call the third interior angle δ.Then α ≥ 90 and β ≥ 90. Henceσ(△ABC) = α + β + δ≥ 180 + δBut by the Saccheri-Legendre Theorem (theorem 30.6) σ(△ABC) ≤ 180.Hence δ = 0, which is impossible. Therefore at most one of the angles canbe non-accute, making the remaining two acute.167
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Foundations of GeometryLecture Note
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ivCONTENTS30 Triangles in Neutral G
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2 SECTION 1. PREFACEare enrolled in
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8 SECTION 2. NCTMTechnology. Techno
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12 SECTION 3. CA STANDARDIn 2005 Ca
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16 SECTION 3. CA STANDARDCalifornia
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34 SECTION 8. HILBERT’S AXIOMSiom
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36 SECTION 8. HILBERT’S AXIOMS«
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38 SECTION 9. BIRKHOFF/MACLANE AXIO
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40 SECTION 9. BIRKHOFF/MACLANE AXIO
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42 SECTION 10. THE SMSG AXIOMSP is
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46 SECTION 11. THE UCSMP AXIOMSin s
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50 SECTION 12. VENEMA’S AXIOMS3.
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52 SECTION 12. VENEMA’S AXIOMS«
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54 SECTION 13. INCIDENCE GEOMETRYFi
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56 SECTION 13. INCIDENCE GEOMETRYEx
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58 SECTION 13. INCIDENCE GEOMETRYFi
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60 SECTION 13. INCIDENCE GEOMETRY«
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62 SECTION 14. BETWEENNESSFigure 14
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64 SECTION 14. BETWEENNESSTheorem 1
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66 SECTION 14. BETWEENNESS(a) To sh
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68 SECTION 14. BETWEENNESSThe follo
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70 SECTION 14. BETWEENNESSExample 1
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72 SECTION 14. BETWEENNESSorf(A) >
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74 SECTION 14. BETWEENNESS« CC BY-
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76 SECTION 15. THE PLANE SEPARATION
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78 SECTION 15. THE PLANE SEPARATION
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80 SECTION 16. ANGLESFigure 16.1: T
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82 SECTION 16. ANGLESCorollary 16.6
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84 SECTION 16. ANGLESFigure 16.6: I
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86 SECTION 16. ANGLESD cannot be in
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88 SECTION 17. THE CROSSBAR THEOREM
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90 SECTION 17. THE CROSSBAR THEOREM
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92 SECTION 18. LINEAR PAIRSFigure 1
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94 SECTION 18. LINEAR PAIRSγ + ∠
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96 SECTION 19. ANGLE BISECTORSwith
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98 SECTION 19. ANGLE BISECTORSAngle
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100 SECTION 20. THE CONTINUITY AXIO
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102 SECTION 20. THE CONTINUITY AXIO
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104 SECTION 21. SIDE-ANGLE-SIDEBirk
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106 SECTION 21. SIDE-ANGLE-SIDEFigu
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108 SECTION 22. NEUTRAL GEOMETRYEll
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110 SECTION 22. NEUTRAL GEOMETRY«
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112 SECTION 23. ANGLE-SIDE-ANGLEFig
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114 SECTION 23. ANGLE-SIDE-ANGLE«
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216 SECTION 40. CIRCLES AND TRIANGL
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218 SECTION 40. CIRCLES AND TRIANGL
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238 SECTION 43. INDIANA BILL 246the
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240 SECTION 44. ESTIMATING πFigure
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242 SECTION 44. ESTIMATING πso tha
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246 SECTION 44. ESTIMATING πTable
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248 SECTION 44. ESTIMATING πnums =
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252 SECTION 45. EUCLIDEAN CONSTRUCT
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264 SECTION 46. HYPERBOLIC GEOMETRY
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272SECTION 47.PERPENDICULAR LINES I
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274SECTION 47.PERPENDICULAR LINES I
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276 SECTION 48. PARALLEL LINES IN H
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298 SECTION 51. THE POINCARE DISK M
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306 SECTION 52. ARC LENGTHAs we see
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308 SECTION 52. ARC LENGTHProof. Su
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310 SECTION 52. ARC LENGTHThus the
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312 SECTION 52. ARC LENGTHMeasuring
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314 SECTION 53. SPHERICAL GEOMETRYW
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318 SECTION 53. SPHERICAL GEOMETRYP
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320 APPENDIX A. SYMBOLS USEDAppendi
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322 APPENDIX A. SYMBOLS USEDTechnol
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Plane Geometry - Bruce E. Shapiro

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