Plane Geometry - Bruce E. Shapiro

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268 SECTION 46. HYPERBOLIC GEOMETRYThen we can choose a point B ′ ∈ AB and C ′ ∈ AC such that AC ′ = DFand AB ′ = DE.By SAS △AB ′ C ′ ∼ = △DEF . Hence∠AB ′ C ′ = ∠E = ∠B∠AC ′ B ′ = ∠F = ∠CSince ∠BB ′ C ′ + ∠AB ′ C ′ = 180 then∠BB ′ C ′ + ∠B = 180Similarly, since ∠B ′ C ′ A + ∠B ′ C ′ C = 180 then∠B ′ C ′ C + ∠C = 180Hence σ(□BB ′ C ′ C) = 360, which contradicts theorem 46.5 which says thatthe angle sum must be strictly less than 360.Hence our RAA hypothesis, which is that hte triangles are not congruent,is false.Theorem 46.15 Let □ABCD and □A ′ B ′ C ′ D ′ be Saccheri quadrilateralswith equal defect and congruent summits. Then □ABCD ∼ = □A ′ B ′ C ′ D ′ .Proof. Let □ABCD and □A ′ B ′ C ′ D ′ be Saccheri quadrilaterals such thatδ(□ABCD) = δ(□A ′ B ′ C ′ D ′ )CD ∼ = C ′ D ′Since σ(□) = 360 − δ(□), for any quadrilateral,σ(□ABCD) = σ(□ ′ B ′ C ′ D ′ )Since the summit angles in any Saccheri Quadrilateral are congruent∠C = ∠D, ∠C ′ = ∠D ′Since both quadrilaterals have the same measure and their base angles areall right angles, then the sum of the summit angles are equal.∠C + ∠D = ∠C ′ + ∠D ′=⇒ 2∠C = 2∠C ′=⇒ ∠C = ∠D = ∠C ′ = ∠D ′« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
SECTION 46. HYPERBOLIC GEOMETRY 269Figure 46.5: If the summits of two Saccheri quadrilaterals with equal defectare congruent, then the two quadrilaterals are congruent.Chose points F ∈ −→ DA and G ∈−→ CB such that DF = D ′ A ′ = CG.By SAS, △F DC ∼ = △A ′ D ′ C ′ . Henceand therefore∠F CD = ∠A ′ C ′ D ′=⇒ ∠GCF = ∠C − ∠F CDBy SAS, △GCF ∼ = △B ′ C ′ A ′ , Hence= ∠C ′ − ∠A ′ C ′ D ′= ∠B ′ C ′ A ′∠G = ∠B ′ = 90Similarly,∠F = ∠A = 90Hence □F GCD ∼ = □A ′ B ′ C ′ D ′ .Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.
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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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4 SECTION 1. PREFACEre-learning it
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6 SECTION 2. NCTM3. To guide in the
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8 SECTION 2. NCTMTechnology. Techno
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10 SECTION 2. NCTM« CC BY-NC-ND 3.
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12 SECTION 3. CA STANDARDIn 2005 Ca
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16 SECTION 3. CA STANDARDCalifornia
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18 SECTION 4. COMMON COREAt all lev
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20 SECTION 5. LOGIC AND PROOFour th
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24 SECTION 5. LOGIC AND PROOF“I
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26 SECTION 6. REAL NUMBERSThe inter
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28 SECTION 6. REAL NUMBERS“The Su
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30 SECTION 7. EUCLID’S ELEMENTSEu
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32 SECTION 7. EUCLID’S ELEMENTSFi
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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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44 SECTION 10. THE SMSG AXIOMS« CC
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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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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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116 SECTION 24. EXTERIOR ANGLESTheo
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118 SECTION 24. EXTERIOR ANGLES« C
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120 SECTION 25. ANGLE-ANGLE-SIDEBy
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122 SECTION 26. SIDE-SIDE-SIDECase
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124 SECTION 26. SIDE-SIDE-SIDEBy si
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126 SECTION 27. SCALENE AND TRIANGL
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128 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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136 SECTION 29. TRANSVERSALSCorolla
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138 SECTION 30. TRIANGLES IN NEUTRA
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146 SECTION 31. QUADRILATERALS IN N
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156 SECTION 32. THE EUCLIDEAN PARAL
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168 SECTION 33. RECTANGLESLemma 33.
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170 SECTION 33. RECTANGLESFigure 33
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172 SECTION 33. RECTANGLES« CC BY-
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174 SECTION 34. THE PARALLEL PROJEC
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176 SECTION 34. THE PARALLEL PROJEC
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178 SECTION 35. SIMILAR TRIANGLESBy
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180 SECTION 35. SIMILAR TRIANGLESCo
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182 SECTION 36. TRIANGLE CENTERSFig
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184 SECTION 36. TRIANGLE CENTERSThe
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186 SECTION 36. TRIANGLE CENTERSCen
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188 SECTION 37. AREADefinition 37.5
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190 SECTION 37. AREAToday’s Lesso
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192 SECTION 38. THE PYTHAGOREAN THE
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202 SECTION 39. CIRCLESFigure 39.1:
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208 SECTION 39. CIRCLESTheorem 39.1
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210 SECTION 39. CIRCLESLet D be the
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212 SECTION 39. CIRCLES« CC BY-NC-
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214 SECTION 40. CIRCLES AND TRIANGL
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216 SECTION 40. CIRCLES AND TRIANGL
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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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