Plane Geometry - Bruce E. Shapiro

More documents

Recommendations

Info

$NAME: Math 103L: Exercise on Functions ... - Bruce E. Shapiro$

$Introducing Latex - Bruce E. Shapiro$

$Applied Differential Equations, Math 280 at CSUN - Bruce E. Shapiro$

$Math 103 Section 1.2: Linear Equations and ... - Bruce E. Shapiro$

112 SECTION 23. ANGLE-SIDE-ANGLEFigure 23.2: Illustration of proof of ASA (theorem 23.1).Since ∠ABC ∼ = ∠DEF (given) and ∠DEF ∼ = ∠ABG (by (23.1)), weconclude that ∠ABC ∼ = ∠ABG.By the angle construction postulate the rays −→ −→BG = BC (because theycorrespond to the same angle measured from −→ BA).Hence point G ∈ −→−→←→BC, and we see that ray BC intersects line AC at bothC and G. Hence line ←→ BC intersects ←→ AC at these same two points.But distinct non-parallel lines can only intersect in one point, so C = G.Hence △ABC ∼ = △ABG ∼ = △DEF .Corollary 23.2 (Converse of Isosceles Triangle Theorem) If △ABCis a triangle that satisfies ∠ACB ∼ = ∠ABC then AB = AC.Proof. Since BC ∼ = CB then by SAS △ABC ∼ = △BAC. Hence AB ∼ =AC.Theorem 23.3 Congruent Triangle Construction Theorem. It ispossible to construct a congruent copy of any triangle on a base, i.e., forany triangle △ABC, if DE ∼ = AB and H is either halfplane of ←→ DE thenthere exists a unique point F ∈ H such that △ABC ∼ = △DEF .Proof. Existence. Suppose that DE ∼ = AB, and choose a side H of AB.Let G ∈ H be chosen so that ∠ABC = ∠DEG (angle construction postulate).Choose F ∈ −→ EG such that EF = BC (ruler postulate).Hence by SAS, △ABC ∼ = △DEF« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
SECTION 23. ANGLE-SIDE-ANGLE 113Figure 23.3: Illustration of proof of existence of construction of triangle ona base, theorem 23.3.Figure 23.4: Illustration of proof of uniqueness of construction of a triangleon a base, theorem 23.3.Uniqueness. To show that F is unique, suppose that there is a second pointH such that △DEF ∼ = △DEH.Since ∠DEF = ∠DEH, H ∈ −→ EG (uniqueness of ray constructed by angleconstruction postulate).Since △DEF ∼ = △DEH, then EF = EH. Hence H = G, because of theuniqueness part of the ruler postulate.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.
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 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
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
Page 52 and 53:
48 SECTION 11. THE UCSMP AXIOMS« C
Page 54 and 55:
50 SECTION 12. VENEMA’S AXIOMS3.
Page 56 and 57:
52 SECTION 12. VENEMA’S AXIOMS«
Page 58 and 59:
54 SECTION 13. INCIDENCE GEOMETRYFi
Page 60 and 61:
56 SECTION 13. INCIDENCE GEOMETRYEx
Page 62 and 63:
58 SECTION 13. INCIDENCE GEOMETRYFi
Page 64 and 65:
60 SECTION 13. INCIDENCE GEOMETRY«
Page 66 and 67: 62 SECTION 14. BETWEENNESSFigure 14
Page 68 and 69: 64 SECTION 14. BETWEENNESSTheorem 1
Page 70 and 71: 66 SECTION 14. BETWEENNESS(a) To sh
Page 72 and 73: 68 SECTION 14. BETWEENNESSThe follo
Page 74 and 75: 70 SECTION 14. BETWEENNESSExample 1
Page 76 and 77: 72 SECTION 14. BETWEENNESSorf(A) >
Page 78 and 79: 74 SECTION 14. BETWEENNESS« CC BY-
Page 80 and 81: 76 SECTION 15. THE PLANE SEPARATION
Page 82 and 83: 78 SECTION 15. THE PLANE SEPARATION
Page 84 and 85: 80 SECTION 16. ANGLESFigure 16.1: T
Page 86 and 87: 82 SECTION 16. ANGLESCorollary 16.6
Page 88 and 89: 84 SECTION 16. ANGLESFigure 16.6: I
Page 90 and 91: 86 SECTION 16. ANGLESD cannot be in
Page 92 and 93: 88 SECTION 17. THE CROSSBAR THEOREM
Page 94 and 95: 90 SECTION 17. THE CROSSBAR THEOREM
Page 96 and 97: 92 SECTION 18. LINEAR PAIRSFigure 1
Page 98 and 99: 94 SECTION 18. LINEAR PAIRSγ + ∠
Page 100 and 101: 96 SECTION 19. ANGLE BISECTORSwith
Page 102 and 103: 98 SECTION 19. ANGLE BISECTORSAngle
Page 104 and 105: 100 SECTION 20. THE CONTINUITY AXIO
Page 106 and 107: 102 SECTION 20. THE CONTINUITY AXIO
Page 108 and 109: 104 SECTION 21. SIDE-ANGLE-SIDEBirk
Page 110 and 111: 106 SECTION 21. SIDE-ANGLE-SIDEFigu
Page 112 and 113: 108 SECTION 22. NEUTRAL GEOMETRYEll
Page 114 and 115: 110 SECTION 22. NEUTRAL GEOMETRY«
Page 118 and 119: 114 SECTION 23. ANGLE-SIDE-ANGLE«
Page 120 and 121: 116 SECTION 24. EXTERIOR ANGLESTheo
Page 122 and 123: 118 SECTION 24. EXTERIOR ANGLES« C
Page 124 and 125: 120 SECTION 25. ANGLE-ANGLE-SIDEBy
Page 126 and 127: 122 SECTION 26. SIDE-SIDE-SIDECase
Page 128 and 129: 124 SECTION 26. SIDE-SIDE-SIDEBy si
Page 130 and 131: 126 SECTION 27. SCALENE AND TRIANGL
Page 132 and 133: 128 SECTION 27. SCALENE AND TRIANGL
Page 134 and 135: 130 SECTION 28. CHARACTERIZATION OF
Page 136 and 137: 132 SECTION 28. CHARACTERIZATION OF
Page 138 and 139: 134 SECTION 29. TRANSVERSALSFigure
Page 140 and 141: 136 SECTION 29. TRANSVERSALSCorolla
Page 142 and 143: 138 SECTION 30. TRIANGLES IN NEUTRA
Page 150 and 151: 146 SECTION 31. QUADRILATERALS IN N
Page 160 and 161: 156 SECTION 32. THE EUCLIDEAN PARAL
Page 166 and 167:
162 SECTION 32. THE EUCLIDEAN PARAL
Page 168 and 169:
Page 170 and 171:
Page 172 and 173:
168 SECTION 33. RECTANGLESLemma 33.
Page 174 and 175:
170 SECTION 33. RECTANGLESFigure 33
Page 176 and 177:
172 SECTION 33. RECTANGLES« CC BY-
Page 178 and 179:
174 SECTION 34. THE PARALLEL PROJEC
Page 180 and 181:
176 SECTION 34. THE PARALLEL PROJEC
Page 182 and 183:
178 SECTION 35. SIMILAR TRIANGLESBy
Page 184 and 185:
180 SECTION 35. SIMILAR TRIANGLESCo
Page 186 and 187:
182 SECTION 36. TRIANGLE CENTERSFig
Page 188 and 189:
184 SECTION 36. TRIANGLE CENTERSThe
Page 190 and 191:
186 SECTION 36. TRIANGLE CENTERSCen
Page 192 and 193:
188 SECTION 37. AREADefinition 37.5
Page 194 and 195:
190 SECTION 37. AREAToday’s Lesso
Page 196 and 197:
192 SECTION 38. THE PYTHAGOREAN THE
Page 198 and 199:
Page 200 and 201:
Page 202 and 203:
Page 204 and 205:
Page 206 and 207:
202 SECTION 39. CIRCLESFigure 39.1:
Page 208 and 209:
Page 210 and 211:
Page 212 and 213:
208 SECTION 39. CIRCLESTheorem 39.1
Page 214 and 215:
210 SECTION 39. CIRCLESLet D be the
Page 216 and 217:
212 SECTION 39. CIRCLES« CC BY-NC-
Page 218 and 219:
214 SECTION 40. CIRCLES AND TRIANGL
Page 220 and 221:
Page 222 and 223:
Page 224 and 225:
220 SECTION 41. EUCLIDEAN CIRCLESFi
Page 226 and 227:
Page 228 and 229:
Page 230 and 231:
Page 232 and 233:
228 SECTION 41. EUCLIDEAN CIRCLESBy
Page 234 and 235:
230 SECTION 42. AREA AND CIRCUMFERE
Page 236 and 237:
Page 238 and 239:
Page 240 and 241:
Page 242 and 243:
238 SECTION 43. INDIANA BILL 246the
Page 244 and 245:
240 SECTION 44. ESTIMATING πFigure
Page 246 and 247:
242 SECTION 44. ESTIMATING πso tha
Page 248 and 249:
244 SECTION 44. ESTIMATING πn π n
Page 250 and 251:
246 SECTION 44. ESTIMATING πTable
Page 252 and 253:
248 SECTION 44. ESTIMATING πnums =
Page 254 and 255:
250 SECTION 44. ESTIMATING π« CC
Page 256 and 257:
252 SECTION 45. EUCLIDEAN CONSTRUCT
Page 258 and 259:
Page 260 and 261:
Page 262 and 263:
Page 264 and 265:
Page 266 and 267:
Page 268 and 269:
264 SECTION 46. HYPERBOLIC GEOMETRY
Page 270 and 271:
Page 272 and 273:
Page 274 and 275:
Page 276 and 277:
272SECTION 47.PERPENDICULAR LINES I
Page 278 and 279:
274SECTION 47.PERPENDICULAR LINES I
Page 280 and 281:
276 SECTION 48. PARALLEL LINES IN H
Page 282 and 283:
Page 284 and 285:
Page 286 and 287:
Page 288 and 289:
284 SECTION 49. TRIANGLES IN HYPERB
Page 290 and 291:
Page 292 and 293:
Page 294 and 295:
290 SECTION 50. AREA IN HYPERBOLIC
Page 296 and 297:
Page 298 and 299:
Page 300 and 301:
Page 302 and 303:
298 SECTION 51. THE POINCARE DISK M
Page 304 and 305:
Page 306 and 307:
Page 308 and 309:
Page 310 and 311:
306 SECTION 52. ARC LENGTHAs we see
Page 312 and 313:
308 SECTION 52. ARC LENGTHProof. Su
Page 314 and 315:
310 SECTION 52. ARC LENGTHThus the
Page 316 and 317:
312 SECTION 52. ARC LENGTHMeasuring
Page 318 and 319:
314 SECTION 53. SPHERICAL GEOMETRYW
Page 320 and 321:
316 SECTION 53. SPHERICAL GEOMETRYF
Page 322 and 323:
318 SECTION 53. SPHERICAL GEOMETRYP
Page 324 and 325:
320 APPENDIX A. SYMBOLS USEDAppendi
Page 326 and 327:
322 APPENDIX A. SYMBOLS USEDTechnol
show all

Plane Geometry - Bruce E. Shapiro

Create successful ePaper yourself

Delete template?

Save as template?