466 Index is cyclic and tangential, 271 regular quadrilateral, 182, 183, 194 relation, mathematical, 24, 384 relative consistency, 32 remote interior angle in a triangle, 113 in an asymptotic triangle, 351 rhombus, 175, 182 is a parallelogram, 182 Riemann, Bernhard, 11 Riemannian geometry, 11 right angle Euclid’s definition, 2 in plane geometry, 88, 92 in the Cartesian plane, 126 right triangle, 105, 229–243 area of, 207 hypotenuse, 105 isosceles, 233 leg, 105 proportion theorem, 237 similarity theorem, 236 rigid motion, 441 root of a polynomial, 314 rotation, 442, 449 rule of inference, 408 ruler flipping lemma, 58 ruler placement theorem, 58 ruler postulate, 57 SMSG, 21, 379 ruler sliding lemma, 58 SA congruence theorem, 351 SA inequality for asymptotic triangles, 353 SAAAS congruence, 197 SAASS congruence, 197 Saccheri quadrilateral, 340, 342, 343 Saccheri’s repugnant theorem, 365 Saccheri, Giovanni, 9, 10, 326, 341, 367 Saccheri–Legendre theorem, 327 for convex polygons, 330 SAS congruence theorem of Euclid, 107 SAS postulate, 108 SAS similarity postulate, 335 SAS similarity theorem, 218 SASAS congruence, 179 SASSS congruence, 197 scale factor, 214 scalene inequality, 115 for quadrilaterals, 342 scalene triangle, 104 scaling theorem circumference, 288 diagonal, 226 height, 226 perimeter, 226 quadrilateral area, 226 triangle area, 226 School Mathematics Study Group, 21, 379 secant line, 249 sector of a circle, 294 area of, 294 segment, 64 contains infinitely many points, 70 coordinate representation of, 67 is convex, 80 lying on a side of a line, 149 segment addition theorem, 66 segment construction theorem, 74 segment cutoff theorem, Euclid’s, 74 segment extension theorem, 65 segment subtraction theorem, 66 semicircle, 254 semiparallel criterion for convexity, 161 of quadrilaterals, 176 semiparallel segments, 160 semiperimeter, 243 set, 423 defined by a list, 424 defined by specification, 424 membership criterion, 423 set-builder notation, 425 set difference, 425 set equality, 424 set postulate, 55 set theory, 423–431 setting out (in Greek proofs), 6 side included between two angles, 104 of a line, 76 of a polygon, 156 of a triangle, 103 of an angle, 83 opposite an angle, 104 side-angle-side congruence theorem of Euclid, 107 side-angle-side postulate, 108 side-side-side congruence theorem, 107 side-splitter theorem, 215 converse, 219 similar polygons, 213 similar triangle construction theorem, 217 similarity theorem

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