Plane Geometry - Bruce E. Shapiro

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16 SECTION 3. CA STANDARDCalifornia <strong>Geometry</strong> Content Standards (Continued)19. Students use trigonometric functions to solve for an unknown length of a sideof a right triangle, given an angle and a length of a side.20. Students know and are able to use angle and side relationships in problemswith special right triangles, such as 30 ◦ , 60 ◦ , and 90 ◦ triangles and 45 ◦ , 45 ◦ ,and 90 ◦ triangles.21. Students prove and solve problems regarding relationships among chords, secants,tangents, inscribed angles, and inscribed and circumscribed polygons ofcircles.22. Students know the effect of rigid motions on figures in the coordinateThe Dragon of Proof« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
Section 4Common CoreThe Common Core Standards Initiative has been an attempt to unify thevariously confusing and conflicting state curricula and to bring them intoalignment with one another. It is coordinated by the National GovernorsAssociation for Best Practices (NGA) and the Council of Chief StateSchools Officers (CCSSO). These standards 1 attempt to defined the knowledgeand skills that students should acquire during their K-12 education,in order to survive basic college curricula once they graduate.Common Core Standards: Goals1. Are aligned with college and work expectations;2. Are clear, understandable and consistent;3. Include rigorous content and application of knowledge through high-order skills;4. Build upon strengths and lessons of current state standards;5. Are informed by other top performing countries, so that all students are prepared tosucceed in our global economy and society; and6. Are evidence-based.The mathematics standards are progressive, with each level keyed to thenext, so that upon successful completion at any given level, the student isprepared for the subsequent level. They provide for a strong foundation innumbers and basic operations (addition, subtraction, multiplication, division)and fractions at the elementary school levels; geometry, algebra, andprobability and statistics in middle school; and and am emphasis on mathematicalmodeling (applying math to practical problems) in high school.1 see http://www.corestandards.org/about-the-standards17
Page 1: Foundations of GeometryLecture Note
Page 4 and 5: ivCONTENTS30 Triangles in Neutral G
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 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
Page 36 and 37: 32 SECTION 7. EUCLID’S ELEMENTSFi
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
Page 48 and 49: 44 SECTION 10. THE SMSG AXIOMS« CC
Page 50 and 51: 46 SECTION 11. THE UCSMP AXIOMSin s
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Page 54 and 55: 50 SECTION 12. VENEMA’S AXIOMS3.
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Page 66 and 67: 62 SECTION 14. BETWEENNESSFigure 14
Page 68 and 69: 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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