468 Index three-point plane, 26, 38, 135 three-ring geometry, 31 three-two geometry, 51 transcendental number, 317 transformation, 108, 441 transitive property of asymptotic rays, 349 transitive property of congruence of angles, 90 of segments, 66 of triangles, 107 transitive property of equality, 393 transitive property of similarity, 214 transitivity of parallelism, 188 transitivity postulate, 323 translation, 442, 449 transversal for a triangle, 221 for two lines, 149 trapezoid, 175 area of, 209 is convex, 177 trapezoid lemma, 178 triangle, 103 acute, 105 area of, 207, 208 constructing, 307 equiangular, 105 equilateral, 104 is convex, 160 is cyclic, 263 is tangential, 268 isosceles, 104 obtuse, 105 right, 105 scalene, 104 triangle area proportion theorem, 209 triangle area scaling theorem, 226 triangle copying theorem, 111 triangle inequality, 116 for real numbers, 437 general, 116 triangle sliding theorem, 208 triangular region, 199 trichotomy law, 437 trigonometry, 238 triple angle formula for cosine, 243 for sine, 243 trisecting an angle, 307, 313–316 truth table, 385 two circles theorem, 252 two-column proof, 405 two-dimensional quantity, 226 two-point equidistance theorem, 152 two-point formula for a line, 34 two-sheeted hyperboloid, 37 two transversals theorem, 218 ultraparallel lines, 355, 360 admit a common perpendicular, 360 distance between, 364 ultraparallel theorem, 360 union, 425 unique line postulate, 56 unique point theorem, 74 unique ray theorem, 89 unique triangle theorem, 111 uniquely defined function, 428 uniqueness, 398 of common perpendicular, 359 proof of, 419 unit circle, 289 universal implication, 412 proof of, 412 universal quantifier, 394, 395 implicit, 397 universal statement, 394 upper bound, 280, 435 least, 280 vacuously true statement, 396 value of a function, 428 variable, 394 free, 394 Varignon’s theorem, 197 vector, 124 components, 124 difference, 125 displacement, 124 sum, 125 vertex of a polygon, 156 of a triangle, 103 of an angle, 83 of an asymptotic triangle, 351 vertex criterion for convexity, 160 vertical angles, 93 vertical angles theorem, 93 converse, 94 vertical line, 33, 125, 136 vertices, see vertex Wallis’s postulate, 324

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