Plane Geometry - Bruce E. Shapiro

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124 SECTION 26. SIDE-SIDE-SIDEBy side-angle-side (AG = AC, α = β, BC = BG), △ABC ∼ = △ABG. Byconstruction △ABG ∼ = △DEF . Hence △ABC ∼ = △DEF .Case 5 (A ∗ B ∗ H). The proof is analogous to Case 4.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
Section 27Scalene and TriangleInequalityTheorem 27.1 (The Scalene Inequality) Let A, B, C be distinct noncollinearpoints. ThenAB > BC ⇐⇒ m(∠ACB) > m(∠BAC) (27.1)Proof. ( =⇒ ) See figure 27.1 (left hand side); the theorem says that sideA is longer than side C if and only if angle c > a. We will use the notationillustrated on the right hand side of the figure. The if we assume AB > BC,we need to show that γ > δ.Figure 27.1: Proof of the Scalene inequality (theorem 27.1).By the ruler postulate there exists a point D ∈ AB such that DB = BC.By the protractor postulate, γ > α.125
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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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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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Page 80 and 81: 76 SECTION 15. THE PLANE SEPARATION
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Page 84 and 85: 80 SECTION 16. ANGLESFigure 16.1: T
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Page 90 and 91: 86 SECTION 16. ANGLESD cannot be in
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Page 96 and 97: 92 SECTION 18. LINEAR PAIRSFigure 1
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Page 104 and 105: 100 SECTION 20. THE CONTINUITY AXIO
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Page 108 and 109: 104 SECTION 21. SIDE-ANGLE-SIDEBirk
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Page 112 and 113: 108 SECTION 22. NEUTRAL GEOMETRYEll
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Page 116 and 117: 112 SECTION 23. ANGLE-SIDE-ANGLEFig
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Page 120 and 121: 116 SECTION 24. EXTERIOR ANGLESTheo
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Page 126 and 127: 122 SECTION 26. SIDE-SIDE-SIDECase
Page 130 and 131: 126 SECTION 27. SCALENE AND TRIANGL
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Page 138 and 139: 134 SECTION 29. TRANSVERSALSFigure
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Page 142 and 143: 138 SECTION 30. TRIANGLES IN NEUTRA
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Page 172 and 173: 168 SECTION 33. RECTANGLESLemma 33.
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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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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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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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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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