Index 465 Proclus’s lemma, 186 Proclus’s postulate, 323 product, Cartesian, 426 projection of a leg, 236 proof, 405 by cases, 414 by contradiction, 416 by contrapositive, 411 by induction, 420 constructive, 417 direct, 410 indirect, 416 justifications in, 406–410 of existence, 417, 418 of existence and uniqueness, 419 stages of writing, 42 structure of, 405 templates for, 405–422 proper angle, 84 proper subset, 424 proper superset, 424 proportion, 213 proportion theorem angle bisector, 219 right triangle, 237 triangle area, 209 proportional, 213 proposition, 6, 24 protractor postulate, 85 Pythagorean identity, 240 Pythagorean postulate, 335 Pythagorean theorem, 229–233, 237 converse, 233 Euclid’s first proof, 230 Euclid’s second proof, 237 Garfield’s proof, 245 proof by decomposition, 232 proof by similar triangles, 237 q.e.d. (quod erat demonstrandum), 7 q.e.f. (quod erat faciendum), 7 quadratic extension of a field, 310 iterated, 311 quadratic polynomial, 314 quadrilateral, 156, 175 area scaling theorem, 226 convex, 175–183 copying theorem, 179 cyclic, 264 tangential, 269 quantifier, 394 existential, 397 multiple, 399 negating, 399 universal, 394, 395 quod erat demonstrandum, 7 quod erat faciendum, 7 radii, see radius radius, 70, 247 range of a function, 428 rating system for proofs, 54 ratio, 213 rational number, 434 rational plane, 136 ray, 71 collinear, 74 coordinate representation of, 72 is convex, 81 lying in the interior of an angle, 95 lying on a side of a line, 78, 149 opposite, 74, 75 with the same endpoint, 74 real number, 53, 424, 433–439 reciprocal, 434 rectangle, 175, 194, 195 area of, 207 construction theorem, 195, 300 golden, 224 is a parallelogram, 175 is convex, 177 nonexistent in hyperbolic geometry, 337 rectangular region, 199 reflection across a line, 146, 442, 443 reflection postulate, 443 reflex angle, 88 reflex measure of an angle, 87 reflexive property of congruence of angles, 90 of segments, 66 reflexive property of equality, 393 region circular, 290 determined by a polygon, 164, 199 polygonal, 199, 200 regular heptagon, 308 regular polygon, 159, 271, 272 angle measures of, 192 center of, 271 central angle of, 271 circumscribed, 273 constructing, 317–319 fifteen-sided, 307 inscribed in a circle, 272
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