456 Index in the Poincar´ disk, 133 interior, 173 of a regular polygon, 192 properties of, 86 reflex, 87 standard, 87 angle measure postulate, 85 angle measurement postulate (SMSG), 21, 380 angle of parallelism, 367 angle-side-longer-side congruence theorem, 121 angle subtraction theorem, 90 angle sum, 189 of an asymptotic triangle, 351 of two angles of a triangle, 114 angle-sum postulate, 325 weak, 333 angle-sum theorem for asymptotic triangles, 352 for convex polygons, 191 for convex quadrilaterals, 194 for general polygons, 193 for triangles, 189, 325 hyperbolic, 338 arbitrary, 412 arc, 254 inscribed, 257 intercepted, 257 major, 254 minor, 254 semicircle, 254 arc addition theorem, 261 arc length, 293 arc measure, 257 arccosine, 125 Archimedean property, 438 Archimedes’ theorem, 292 area, 199–210 of a circular region, 291–293 of a parallelogram, 209 of a polygon, 200 of a rectangle, 207 of a right triangle, 207 of a sector, 294 of a square, 206 of a trapezoid, 209 of a triangle, 208 area addition property, 200 area congruence property, 200 area function, 200 area paradox, 211 area postulate Euclidean, 200 hyperbolic, 371 is not independent, 209 area scaling theorem quadrilateral, 226 triangle, 226 Aristotle, 363 Aristotle’s lemma, 363 ASA congruence theorem, 108 ASAAS congruence, 197 ASASA congruence, 197 ASASS congruence, 197 AsS congruence theorem, 121 ASS nontheorem, 120, 122 ASSAS congruence, 197 asymptotic rays, 344 determine parallel lines, 344 endpoint independence, 348 existence and uniqueness, 345 symmetry property, 347 transitive property, 349 asymptotic triangle, 350 AA congruence theorem, 353 angle sum of, 351 angle-sum theorem, 352 copying theorem, 352 defect addition theorem, 353 defect of, 351 exterior angle inequality, 353 exterior angle of, 351 Pasch’s theorem for, 353 remote interior angle of, 351 SA congruence theorem, 351 SA inequality, 353 asymptotically parallel lines, 355 distance between, 365 existence and uniqueness, 357 atomic statement, 384 axiom, 13, 23 of incidence geometry, 25 axiomatic method, 23 axiomatic system, 13, 23 base of a parallelogram, 209 of a Saccheri quadrilateral, 341 of a trapezoid, 209 of a triangle, 208 of an isosceles triangle, 105 base angles, 105

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