Plane Geometry - Bruce E. Shapiro

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282 SECTION 48. PARALLEL LINES IN HYPERBOLIC GEOMETRYTheorem 48.16 If l ‖ m are parallel lines that admit a common perpendicular,the for every d 0 > 0 there exists a point P ∈ m such that d(P, l) > d 0 ;and P may be chosen to line on either side of the common perpendicular.Proof. (omitted)Theorem 48.17 (Transitivity of Limiting Parallels) If l is asymptoticallyparallel to m in the direction −→ AB and l is asymptotically parallel ton in the direction −→ AB, then either m = n or m is asymptotically parallelto n.Proof. (omitted)Theorem 48.18 If l ‖ m then either l and m admit a common perpendicularor l and m are asymptotically parallel.We have already shown that the critical function is decreasing. In fact, itis strictly decreasing.Theorem 48.19 The critical function κ(x) is strictly decreasing.Proof. (See Venema.)Theorem 48.20 If l and m are asymptotically parallel lines then for everyɛ > 0 there exists a point T ∈ m such that d(T, l) < ɛ.Proof. (See Venema.)Theorem 48.21 limx→∞ κ(x) = 0Proof. (See Venema.)Theorem 48.22 limx→0 + κ(x) = 90Proof. (See Venema.)Theorem 48.23 The critical funnction κ(x) is onto.Proof. (Exercise.)Theorem 48.24 The critical functionis continuous.Proof. (Exercise.)κ(x) : (0, ∞) ↦→ (0, 90)« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
Section 49Triangles in Hyperbolic<strong>Geometry</strong>Theorem 49.1 For every ɛ > 0 there exists an isosceles triangle △BP Cwith σ(△BP C) < ɛ and δ(△BP C) > 180 − ɛ.Proof. Let ɛ > 0 be given.Since lim x→∞ κ(x) = 0 there exists some y such that ∀a > y, κ(a) < ɛ/4.Choose one such a.Let △P AB be a triangle with right angle at A and both legs of length a.Since −→−→P B intersects AB then µ(∠AP B) < K(a).Since ∠ABP = ∠AP B,andσ△BP C = µ(∠ACP ) + µ(∠ABP )+ 2µ(∠AP B)= 4µ(∠AP B)< 4κ(a)< ɛδ(△BP C) = 180 − σ(△BP C)< 180 − ɛTheorem 49.2 For every ɛ > 0 there is a right triangle △ABC such thatσ(△ABC) > 180 − ɛ and δ(△ABC) < ɛ.283
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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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38 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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48 SECTION 11. THE UCSMP AXIOMS« C
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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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68 SECTION 14. BETWEENNESSThe follo
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70 SECTION 14. BETWEENNESSExample 1
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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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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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116 SECTION 24. EXTERIOR ANGLESTheo
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120 SECTION 25. ANGLE-ANGLE-SIDEBy
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126 SECTION 27. SCALENE AND TRIANGL
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130 SECTION 28. CHARACTERIZATION OF
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132 SECTION 28. CHARACTERIZATION OF
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134 SECTION 29. TRANSVERSALSFigure
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138 SECTION 30. TRIANGLES IN NEUTRA
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170 SECTION 33. RECTANGLESFigure 33
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174 SECTION 34. THE PARALLEL PROJEC
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178 SECTION 35. SIMILAR TRIANGLESBy
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182 SECTION 36. TRIANGLE CENTERSFig
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188 SECTION 37. AREADefinition 37.5
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Page 280 and 281: 276 SECTION 48. PARALLEL LINES IN H
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Plane Geometry - Bruce E. Shapiro

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