AMSCO'S Geometry. New York - Rye High School

638 IndexDefinition(s) cont.using, in proofs, 141–142writing as conditionals, 98Degree measureof angle, 16of arc, 537–538DeMorgan, Augustus, 34DeMorgan’s Laws, 34Descartes, René, 209, 290Detachment, law of, 75, 101, 105Diagonalof polygon, 368of quadrilateral, 380Diagram(s)tree, 42using, in geometry, 26–27Diameter, 543Dihedral angle, 424Dilation(s), 495–499in coordinate plane, 247–248defined, 247preservation of angle measure under,496–497preservation of collinearity under, 498preservation of midpoint under, 497Direct isometry, 252–253Direct proof, 105–106Disjunct, 48Disjunction, 48–50, 76Disjunctive inference, law of, 76–78Distance,between two parallel lines, 383between two planes, 437between two points, 7–8from a point to a line, 20preservation of, under glidereflection, 244preservation of, under pointreflection, 228preservation of, under rotation abouta fixed point, 239preservation of, under translation, 234Distance formula, 521–522, 522Distance postulate, 136Distributive property, 5Divide and average method, 174Division postulate, 124, 270–271D k, 247, 496Domain, 36, 250EEdge of polyhedron, 440Elements (Euclid), 1, 262, 474Endpoint, of ray, 15Enlargement, 496Epicureans, 262Equalityreflexive property of, 110–111symmetric property of, 111transitive property of, 111, 263–264Equation(s)of line, 295–299solving, with biconditionals, 70–71Equiangular triangle, 25Equidistant, 191Equidistant lines in coordinategeometry, 619–622Equilateral triangle(s), 24, 25, 181–183properties of, 183Equivalence relation, 111of congruence, 156–157of similarity, 487–488Eratosthenes, 1Euclid, 1, 93, 134, 262, 328, 379, 535parallel postulate of, 328Euclid Freed of Every Flaw, 379Exclusive or,51Exterior angle(s), 330alternate, 330of polygon, 276–277, 369–371of triangle, 277–279Exterior angle inequality theorem, 277Exterior angle theorem, 349Exterior of angle, 16Exterior of circle, 536External segment, 576Extremes, 476FFace(s) of polyhedron, 440Fermat, Pierre de, 209Fixed point, 214Foot of altitude, 525Foot of perpendicular, 20Formula(s)angles,central, 555inscribed, 555formed by tangents, chords,and secants, 571–572of polygons, 369area of a rectangle, 409circle, 582distance, 522Heron’s, 174lateral area,of cone, 457, 467of cylinder, 454, 467of prism, 442, 467of pyramid, 467midpoint, 304point-slope, 297segments formed by tangents, chords,and secants, 579slope, 292, 297surface area,of cone, 457of cylinder, 454, 467of prism, 442, 467of pyramid, 467of sphere, 462, 467volume,of cone, 457, 467of cylinder, 454, 467of prism, 446, 467of pyramid, 449, 467of sphere, 462, 467Foundations of Geometry (Hilbert), 9345-45-degree right triangle, 517–518Frustum of cone, 459Function(s)defined, 250transformations as, 250–254GGalileo, 419Generalization, 94General quadrilateral, 380Geometric constructions, 196. See alsoConstuctionsGeometric inequalities, 262–285basic inequality postulates, 263–265inequalities involving lengths of thesides of a triangle, 273–274inequalities involving sides andangles of a triangle, 281–284inequality involving an exteriorangle of a triangle, 276–279inequality postulates involvingaddition and subtraction,267–268inequality postulates involvingmultiplication and division,270–271Geometric mean, 478Geometryanalytic, 209coordinate, 209, 290deductive reasoning, 100–103defined, 2definitions as biconditionals, 97–99inductive reasoning, 94–97non-Euclidean, 376spherical, 32proving statements in, 93–130addition and subtractionpostulates, 118–122direct proofs, 105–108indirect proofs, 105–108multiplication and divisionpostulates, 124–126postulates, theorems, and proof,109–115substitution postulate, 115–117solid, 420using diagrams in, 26–27using logic to form proof, 100–103Glide reflection, 243–245Graphing polygons, 212–213Graphs, 4Great circle of sphere, 460, 461HHalf-line, 14–15Heath, Thomas L., 1Height. See also Altitudeof cone, 456of cylinder, 453of prism, 440of pyramid, 449Heron of Alexandria, 174Heron’s formula, 174Hexagon, 367Hidden conditional, 55–57, 98Hilbert, David, 93HL triangle congruence theorem,362–365Hypotenuse, 26Hypotenuse-leg triangle congruencetheorem, 362–365. See alsoHL triangle congruencetheoremHypothesis, 55IIdentityadditive, 5