Index 30-60-90 triangle, 190, 233 intersects interior of a side, 144 36-72-72 triangle, 226 to the base of an isosceles triangle, 145 360 theorem, 96, 97 to the hypotenuse, 144 45-45-90 triangle, 190, 233 to the longest side, 144 60-60-60 triangle, 189 Amtrak model, 29 and (logical conjunction), 385 AA congruence theorem for asymptotic angle, 83 triangles, 353 acute, 88 AA similarity theorem, 216 included between two sides, 104 AAA congruence theorem in hyperbolic inscribed in a semicircle, 257 geometry, 338 inscribed in an arc, 257 AAA construction theorem, 191 obtuse, 88 AAASA congruence, 197, 354 of a polygon, 156 AAS congruence theorem, 119 of a triangle, 103 AASAS congruence, 179 of an asymptotic triangle, 351 ABCD property of rigid motions, 441 on a side of a line, 149 absolute value, 434 opposite a side, 104 acute angle, 88 proper, 84 acute triangle, 105 right, 88 adapted coordinate function, 72 straight, 84 adjacency lemma, 98 zero, 84 adjacent angles, 90, 91 angle addition theorem, 90 adjacent edges of a polygon, 156 angle bisector, 100, 147 adjacent interior angle, 113 angle bisector concurrence theorem, 268 admissible decomposition, 201 angle bisector proportion theorem, 219 algebraic number, 317 angle bisector theorem, 147 all-or-nothing theorem, 333 converse, 149 alternate interior angles, 150 angle construction theorem, 88 alternate interior angles postulate, 323 angle criterion for convexity, 160 alternate interior angles theorem, 150 angle measure, 54, 85 converse, 185, 323 between two lines, 357 altitude concurrence theorem, 269 in taxicab geometry, 138 altitude of a triangle, 144, 208 in the Cartesian plane, 125 455 Downloaded from bookstore.ams.org 456 Index in the Poincare´ disk, 133 area paradox, 211 interior, 173 area postulate of a regular polygon, 192 Euclidean, 200 properties of, 86 hyperbolic, 371 reflex, 87 is not independent, 209 standard, 87 area scaling theorem angle measure postulate, 85 quadrilateral, 226 angle measurement postulate (SMSG), 21, triangle, 226 380 Aristotle, 363 angle of parallelism, 367 Aristotle’s lemma, 363 angle-side-longer-side congruence theorem, ASA congruence theorem, 108 121 ASAAS congruence, 197 angle subtraction theorem, 90 ASASA congruence, 197 angle sum, 189 ASASS congruence, 197 of an asymptotic triangle, 351 AsS congruence theorem, 121 of two angles of a triangle, 114 ASS nontheorem, 120, 122 angle-sum postulate, 325 ASSAS congruence, 197 weak, 333 asymptotic rays, 344 angle-sum theorem determine parallel lines, 344 for asymptotic triangles, 352 endpoint independence, 348 for convex polygons, 191 existence and uniqueness, 345 for convex quadrilaterals, 194 symmetry property, 347 for general polygons, 193 transitive property, 349 for triangles, 189, 325 asymptotic triangle, 350 hyperbolic, 338 AA congruence theorem, 353 arbitrary, 412 angle sum of, 351 arc, 254 angle-sum theorem, 352 inscribed, 257 copying theorem, 352 defect addition theorem, 353 intercepted, 257 defect of, 351 major, 254 exterior angle inequality, 353 minor, 254 exterior angle of, 351 semicircle, 254 Pasch’s theorem for, 353 arc addition theorem, 261 remote interior angle of, 351 arc length, 293 SA congruence theorem, 351 arc measure, 257 SA inequality, 353 arccosine, 125 asymptotically parallel lines, 355 Archimedean property, 438 distance between, 365 Archimedes’ theorem, 292 existence and uniqueness, 357 area, 199–210 atomic statement, 384 of a circular region, 291–293 axiom, 13, 23 of a parallelogram, 209 of incidence geometry, 25 of a polygon, 200 axiomatic method, 23 of a rectangle, 207 axiomatic system, 13, 23 of a right triangle, 207 of a sector, 294 base of a square, 206 of a parallelogram, 209 of a trapezoid, 209 of a Saccheri quadrilateral, 341 of a triangle, 208 of a trapezoid, 209 area addition property, 200 of a triangle, 208 area congruence property, 200 of an isosceles triangle, 105 area function, 200 base angles, 105 Downloaded from bookstore.ams.org Index 457 Beltrami, Eugenio, 11, 35 through three noncollinear points, 264 Beltrami–Klein disk, 35, 41 unit, 289 betweenness circular region, 290 consistency of, 61, 90 area of, 291 in Hilbert’s axioms, 20, 23, 375 circumcenter, 262 of numbers, 59 circumcircle, 262 of points, 60 for a triangle, 263 of rays, 89 circumcircle postulate, 335 symmetry of, 60, 89 circumcircle theorem, 263 vs. betweenness, 99 circumference, 284 vs. interior, 97 approximation by regular polygons, 285 betweenness axiom, Hilbert’s, 61, 375 formula for, 289 betweenness theorem scaling theorem, 288 converse, 61, 117 circumscribed circle, 262 for points, 60 constructing, 305 for rays, 89 for a triangle, 263 biconditional statement, 392 circumscribed polygon, 265 bijective function, 57, 429 is convex, 266 Birkhoff’s axioms, 377 regular, 273 Birkhoff, George D., 20, 53, 377 Clairaut’s postulate, 330 bisect, 68 Clairaut, Alexis, 330 bisector classification of parallels through a point, of an angle, 100, 147, 298 357 perpendicular, 145, 146 closed half-plane, 77 Bolyai, Janos,´ xiii, 10 closed interval, 434 boundary of a simple polygonal region, 199 closest point, 146 braces in set notation, 424 on a line, 147 Cartesian plane, 33–34, 125–132 on a segment, 147 is a model of Euclidean geometry, 132 closure properties of numbers, 435 is a model of incidence geometry, 33 codomain of a function, 428 is a model of neutral geometry, 132 coefficient of a polynomial, 314 Cartesian product, 426 collection (synonym for set), 423 cases, proof by, 414 collinear points categorical axiomatic system, 132, 243 in incidence geometry, 25 Euclidean geometry, 132, 243 in plane geometry, 55 center collinear rays, 74 of a circle, 70, 247 common notions of a regular polygon, 271 for angles, 90 central angle for segments, 66 of a circle, 257 of Euclid, 4 of a regular polygon, 271 common perpendicular, 151, 187, 359, 360 centroid, 224 uniqueness, 359 Ceva’s theorem, 223 common perpendicular theorem, 151 Ceva, Giovanni, 223 converse, 187 cevian, 223 compass and straightedge, 3, 295, 296 chord complementary angles, 91 of a circle, 35, 247, 250 components of a polygon, 161, 162 of a vector, 124 circle, 70, 247 of an ordered pair, 426 area of, 293 composition of rigid motions, 449 circumference of, 289 compound statement, 385 Downloaded from bookstore.ams.org 458 Index concave polygon, 158 constructing a perpendicular, 101 concave vertex, 172 constructing a rectangle, 195 concentric circles, 247, 248 constructing a square, 196 conclusion construction in Greek proofs, 7 compass and straightedge, 296–319 of a rule of inference, 408 in Greek proofs, 3, 6, 7 of an implication, 388 constructive proof, 417 concurrence theorem contains (a point) altitude, 269 in incidence geometry, 24 angle bisector, 268 in plane geometry, 55 median, 224 contrapositive, 391 perpendicular bisector, 264 proof by, 411 concurrent, 223, 262 converse, 391 conditional statement, 388 convex decomposition lemma, 201 congruence convex polygon, 157–166 in Hilbert’s axioms, 20, 23, 375, 376 angle criterion, 160 of angles, 88 characterizations of, 173 of asymptotic triangles, 351 diagonal criterion, 176 of convex polygons, 159 interior of, 164, 171 of polygons, 173 semiparallel criterion, 161, 176 of segments, 64 vertex criterion, 160 of simple polygonal regions, 199 convex quadrilateral, 175–183 of triangles, 106 diagonal criterion, 176 via rigid motions, 446 semiparallel criterion, 176 congruent, see congruence convex set, 80 conjugate arcs, 254, 257 intersection of, 80 conjunction, 385 convex vertex, 172 connective, logical, 385 coordinate consecutive angles of a polygon, 156 of a point, 57 consecutive interior angles, 150 ofaray,86 consecutive interior angles theorem, 151 coordinate function converse, 186 adapted to a ray, 72 consecutive vertices of a polygon, 156 for a half-rotation, 86 consistency for a line, 57 of an axiomatic system, 32 starting at a ray, 86 of betweenness of points, 61 coordinate representation of betweenness of rays, 90 ofaray,72 of endpoints of arcs, 256 of a segment, 67 of endpoints of rays, 76 copying theorem of endpoints of segments, 66 asymptotic triangle, 352 of polygon vertices, 157, 159 quadrilateral, 179, 300 of triangle vertices, 103 triangle, 111, 300 of vertices of proper angles, 84 corollary, 6, 24 relative, 32 correspondence, one-to-one, 429 constant of proportionality, 214 corresponding angles, 150 constructible angle, 313 corresponding angles theorem, 151 constructible circle, 308 converse, 186 constructible line, 308 cosine, 238, 239 constructible number, 308 is injective, 239 characterization of, 311 cosines, law of, 240 constructible point, 308 cross lemma, 177 Downloaded from bookstore.ams.org Index 459 crossbar theorem, 106 from a point to a segment, 147 cubic polynomial, 314 from a point to a set, 146 cut by a transversal, 149 in plane geometry, 54, 56 cutting off a segment, 74, 297 in taxicab geometry, 138 cyclic polygon, 262, 263 in the Cartesian plane, 125 is convex, 262 in the Poincare´ disk, 132 cyclic quadrilateral, 264, 276 properties of, 59 cyclic triangle theorem, 263 distance postulate, 56 distance-preserving function, 57 dart, 183 distinct (meaning not equal), 25, 393 decagon, 157 divides, 434 decomposition lemma divisible, 434 convex, 201 dodecagon, 157 parallelogram, 203 domain defect of a function, 428 of a polygon, 328, 371 of a variable, 394 of a triangle, 328 dot product, 125 of an asymptotic triangle, 351 double angle formula defect addition theorem, 328 for cosine, 242 for asymptotic triangles, 353 for sine, 242 defect formula, 372 double elliptic geometry, 35