Plane Geometry - Bruce E. Shapiro

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ivCONTENTS30 Triangles in Neutral Geometry . . . . . . . . . . . . . . . . 13731 Quadrilaterals in Neutral Geometry . . . . . . . . . . . . . 14532 The Euclidean Parallel Postulate . . . . . . . . . . . . . . 15533 Rectangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16734 The Parallel Projection Theorem . . . . . . . . . . . . . . 17335 Similar Triangles . . . . . . . . . . . . . . . . . . . . . . . . 17736 Triangle Centers . . . . . . . . . . . . . . . . . . . . . . . . 18137 Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18738 The Pythagorean Theorem . . . . . . . . . . . . . . . . . . 19139 Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20140 Circles and Triangles . . . . . . . . . . . . . . . . . . . . . . 21341 Euclidean Circles . . . . . . . . . . . . . . . . . . . . . . . . 21942 Area and Circumference of Circles . . . . . . . . . . . . . 22943 Indiana Bill 246 . . . . . . . . . . . . . . . . . . . . . . . . . 23744 Estimating π . . . . . . . . . . . . . . . . . . . . . . . . . . . 23945 Euclidean Constructions . . . . . . . . . . . . . . . . . . . . 25146 Hyperbolic Geometry . . . . . . . . . . . . . . . . . . . . . 26347 Perpendicular Lines in Hyperbolic Geometry . . . . . . . 27148 Parallel Lines in Hyperbolic Geometry . . . . . . . . . . . 27549 Triangles in Hyperbolic Geometry . . . . . . . . . . . . . . 28350 Area in Hyperbolic Geometry . . . . . . . . . . . . . . . . 28951 The Poincare Disk Model . . . . . . . . . . . . . . . . . . 29752 Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30553 Spherical Geometry . . . . . . . . . . . . . . . . . . . . . . 313A Symbols Used . . . . . . . . . . . . . . . . . . . . . . . . . . 319Bibliography . . . . . . . . . . . . . . 323« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
Section 1PrefaceReflectionsFoundations of Geometry is a content class for future high school teachers.In most college math classes, the focus is on a logical developmentof mathematical tools and techniques; their underlying theory, derivationsand proofs; and application to the real world or other mathematical subdisciplines.As a content class we will do all this, but we will also makean (albeit limited) attempt to integrate the material in a way that it ispertinent to your future as a an educator.On Pedagogy. As teachers you will need to know the most effective wayto present the material to others. To be effective, you should be clear, concise,and interesting. To be successful your presentation needs to promoteunderstanding and retention.Most of this cannot be learned in the classroom. It requires hard workand lots of practice. I can tell you what I find effective but that might nothelp very much. My students are more mathematically mature than yourswill be - after all, they are college mathematics majors. My students arepeople who like math, want to learn more about it, and (in the your case)are anxious to share that excitement with others. Middle school and highschool is different. So most of what works for me probably won’t work verywell in your classrooms.As you progress towards your California teaching credential you will takea lot of classes in pedagogy, the art or science of teaching. Some of thesewill focus on pedagogy in mathematics and others will not. A few of you1
Page 1: Foundations of GeometryLecture Note
Page 6 and 7: 2 SECTION 1. PREFACEare enrolled in
Page 8 and 9: 4 SECTION 1. PREFACEre-learning it
Page 10 and 11: 6 SECTION 2. NCTM3. To guide in the
Page 12 and 13: 8 SECTION 2. NCTMTechnology. Techno
Page 14 and 15: 10 SECTION 2. NCTM« CC BY-NC-ND 3.
Page 16 and 17: 12 SECTION 3. CA STANDARDIn 2005 Ca
Page 18 and 19: 14 SECTION 3. CA STANDARDThe Califo
Page 20 and 21: 16 SECTION 3. CA STANDARDCalifornia
Page 22 and 23: 18 SECTION 4. COMMON COREAt all lev
Page 24 and 25: 20 SECTION 5. LOGIC AND PROOFour th
Page 26 and 27: 22 SECTION 5. LOGIC AND PROOFRene D
Page 28 and 29: 24 SECTION 5. LOGIC AND PROOF“I
Page 30 and 31: 26 SECTION 6. REAL NUMBERSThe inter
Page 32 and 33: 28 SECTION 6. REAL NUMBERS“The Su
Page 34 and 35: 30 SECTION 7. EUCLID’S ELEMENTSEu
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Page 38 and 39: 34 SECTION 8. HILBERT’S AXIOMSiom
Page 40 and 41: 36 SECTION 8. HILBERT’S AXIOMS«
Page 42 and 43: 38 SECTION 9. BIRKHOFF/MACLANE AXIO
Page 44 and 45: 40 SECTION 9. BIRKHOFF/MACLANE AXIO
Page 46 and 47: 42 SECTION 10. THE SMSG AXIOMSP is
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Page 50 and 51: 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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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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220 SECTION 41. EUCLIDEAN CIRCLESFi
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228 SECTION 41. EUCLIDEAN CIRCLESBy
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230 SECTION 42. AREA AND CIRCUMFERE
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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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244 SECTION 44. ESTIMATING πn π n
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246 SECTION 44. ESTIMATING πTable
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248 SECTION 44. ESTIMATING πnums =
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250 SECTION 44. ESTIMATING π« CC
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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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284 SECTION 49. TRIANGLES IN HYPERB
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290 SECTION 50. AREA IN HYPERBOLIC
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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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316 SECTION 53. SPHERICAL GEOMETRYF
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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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