Plane Geometry - Bruce E. Shapiro

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150 SECTION 31. QUADRILATERALS IN NEUTRAL GEOMETRYTheorem 31.13 If □ABCD is a non-convex quadrilateral then □ACBDis a quadrilateral.Proof. Since □ABCD is a quadrilateral no three of the points A, B, C, Dare collinear.Since □ABCD is a quadrilateral BC ∩ AD = ∅.Since □ABCD is not convex then AC and BD are disjoint (the diagonalsdo not intersect).Thus segments AC, CB, BD, and DA share at most their endpoints.Hence □ACBD is a quadrilateral.Definition 31.14 □ABCD is a Saccheri Quadrilateral if ∠ABC =∠BAD = 90 and AD = BC. Segment AB is called the base and segmentDC is called the summit.Figure 31.5: A Saccheri Quadrilateral. It is not possible to prove that thisfigure is a rectangle using only the axioms of neutral geometry - one mustaccept Euclid’s fifth postulate to do so. Thus we sometimes draw the topand side edges as curves to remind ourselves of this.Theorem 31.15 The diagonals of a Saccheri Quadrilateral are congruent.Proof. Consider triangles △ABD and △ABC. Since BC = AD, AB =AB, and ∠A = 90 = ∠B, they are congruent. Hence BD ∼ = AC.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
SECTION 31. QUADRILATERALS IN NEUTRAL GEOMETRY 151Figure 31.6: The shaded triangles are congruent, hence the correspondingdiagonals of the Saccheri Quadrilateral are congruent (theorem 31.15).DCDCABABTheorem 31.16 The summit angles of a Saccheri Quadrilateral are congruent.Proof. Repeat the argument in the previous proof, but with the upper-halftriangles. The triangles are congruent by SSS - they share the same top;the diagonals are congruent; and the sides are congruent. Hence the cornerangles are congruent.Theorem 31.17 Let □ABCD be a Saccheri quadrilateral. Then the segmentjoining the midpoints of the base and summit is perpendicular to thebase and summit.Proof. Let M be the bisector of AB and N the bisector of CD (see figure31.7).Figure 31.7: Proof of theorem 31.17. The line segment connecting the midpointsof the base and summit of a Saccheri Quadrilateral is perpendicularto both the base and the summit.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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14 SECTION 3. CA STANDARDThe Califo
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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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22 SECTION 5. LOGIC AND PROOFRene D
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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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200 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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