Index triangle, 70, theorem, triangle, triangle, triangle, 70

Transcription

1 Index triangle, 70, theorem, triangle, triangle, triangle, 70 AA similarity theorem, 110 AAA congruence theorem, 154 AAASA, 80 AAS theorem, 46 AASAS, 78 acute angle, 25 acute triangle, 36 additivity of defects, 146 adjacent angles of a quadrilateral, 74 adjacent interior angle, 43 adjacent sides of a quadrilateral, 74 adjacent vertices of a quadrilateral, 74 admissible decomposition, 99 all-or-nothing theorem, 151 alternate interior angles, 64 alternate interior angles theorem, 64 converse, 67, 140 altitude of a triangle, 61, 103 angle, 23 acute, 25 included between two sides, 35 obtuse, 25 of a quadrilateral, 73 of a triangle, 35 opposite a side, 35 right, 25 angle addition theorem, 28 angle bisector, 29, 63 angle bisector proportionality theorem, 112 angle bisector theorem, 63 angle construction theorem, 25 angle measure, 2, 24 between two lines, 175 in the Poincaré disk model, 54 angle measure postulate, 24 angle subtraction theorem, 28 angle sum for a convex quadrilateral, 78 for a polygon, 92 for a triangle, 69 angle-sum postulate, 142 weak, 150 angle-sum theorem for asymptotic triangles, 174 for convex polygons, 84 for convex quadrilaterals, 78 for general polygons, 93 for triangles, 69, 142 hyperbolic, 154 arccosine, 51 area, 2, of a parallelogram, 104 of a polygon, 98 of a polygonal region, 98 of a rectangle, 102 of a right triangle, 103 of a square, 102 of a trapezoid, 105 of a triangle, 103 I-1

2 I-2 INDEX of a unit square, 98 area postulate, 98 independence of, 105 unit, 98 area scaling theorem quadrilateral, 115 triangle, 115 ASA theorem, 39 ASASA, 79 ASS non-theorem, 46, 48 asymptotic rays, 165 determine parallel lines, 166 endpoint independence, 169 existence and uniqueness, 167 symmetry property, 168 transitivity property, 170 asymptotic triangle, 172 angle-sum theorem, 174 exterior angle theorem theorem, 174 SA congruence theorem, 172 asymptotically parallel lines, 167 distance between, 177 existence and uniqueness, 175 base of a parallelogram, 104 of a Saccheri quadrilateral, 156 of a trapezoid, 104 of a triangle, 103 of an isosceles triangle, 36 base angles, 36 Beltrami, Eugenio, 153 betweenness consistency of, 9, 28 of numbers, 8 of points, 8 of rays, 27 symmetry of, 8, 28 vs. betweenness, 33 vs. interior, 32 betweenness theorem converse, 9, 34, 45 for points, 8 for rays, 28 bijective function, 5 Birkhoff, George D., 1 bisector angle, 29, 63 perpendicular, 62 Bolyai, János, 153 boundary of a polygonal region, 97 Cartesian plane, center of a circle, 129 characterizations of convex polygons, 95 chord of a circle, 129 chord theorem, 130 circle, 129 unit, 53 circumcenter, 136 circumcircle, 136 for a triangle, 137 circumcircle theorem, 136 circumscribed circle, 136 for a triangle, 137 circumscribed polygon, 137 Clairaut s axiom, 148 Clairaut, Alexis, 148 closed half-plane, 83 collinear points, 3 collinear rays, 18 common notions for angles, 28 for segments, 12 of Euclid, 11 common perpendicular, 65, 69, 160, 173, 176 uniqueness, 159 common perpendiculars theorem, 65 converse, 69 complementary angles, 26 components of a vector, 50 concave polygon, 83 concave quadrilateral, 75 concave vertex, 86, 92 concentric circles, 129, 130 concurrent, 136 congruent angles, 25 congruent polygons, 92 congruent quadrilaterals, 74 congruent segments, 11

3 INDEX I-3 congruent simple regions, 97 congruent triangles, 37 consecutive interior angles, 64 consecutive interior angles theorem, 65 converse, 67 consistency of betweenness of points, 9 of betweenness of rays, 28 constant of proportionality, 108 constructing a perpendicular, 59 constructing a square, 119 contains (a point), 3 convex polygon, 83, 84 characterizations of, 95 interior of, 93 convex quadrilateral, 75, 76 convex set, 20, 34 convex vertex, 86, 92 coordinate of a point, 5 of a ray, 25 coordinate function for a half-rotation, 25 for a line, 5 coordinate representation of a ray, 16 of a segment, 13 copying a quadrilateral, 78 copying a triangle, 41 corresponding angles, 64 corresponding angles theorem, 65 converse, 67 crossbar theorem, 37 cut by a transversal, 64 cyclic polygon, 136 decagon, 82 defect of a polygon, 146 additivity of, 146 degrees, 24 diagonal interior, 93 of a quadrilateral, 74 of a square, 122 diagonal scaling theorem, 114 diameter of a circle, 129, 130 length of, 130 displacement vector, 50 distance, 2, 4 from a point to a line, 63 in the Poincaré disk model, 54 properties of, 7 distance postulate, 4 dodecagon, 82 dot product, 50 dropping a perpendicular, 59 edge of a quadrilateral, 73 elliptic geometry, 56 endpoint of a ray, 16 of a segment, 11 equiangular polygon, 86 equiangular triangle, 36, 41 equidistant, 159 from two lines, 63 from two points, 62 equidistant lines, 67 are parallel, 67 symmetry of, 68 equilateral polygon, 86 equilateral triangle, 36, 41 equivalent postulates, 71, 139 Euclid s fifth postulate, 140 implied by Euclidean parallel postulate, 70 implies Euclidean parallel postulate, 71 Euclidean axioms, Euclidean geometry, 66 Euclidean parallel postulate, 66, 140 implied by Euclid s fifth postulate, 71 implies Euclid s fifth postulate, 70 even parity, 87 existence of infinitely many points, 7 of parallels, 65

4 I-4 INDEX of three noncollinear points, 19 of two distinct points, 3 existence postulate, 3 exterior of a circle, 129 of a polygon, 87 exterior angle formed by a transversal, 64 of a triangle, 43 exterior angle theorem, 43 for asymptotic triangles, 174 external point for a line, 3 foot of a perpendicular, 61 four right angles theorem, 59 fourth angle of a Lambert quadrilateral, 157 fourth vertex of a Lambert quadrilateral, 157 Gauss, Carl Friedrich, 153 geometric mean, 125 half-plane, 20 closed, 83 is convex, 20 open, 83 half-rotation, 24 height of a parallelogram, 104 of a trapezoid, 104 of a triangle, 103 height scaling theorem, 114 hexagon, 82 Hilbert s parallel postulate, 140 hinge theorem, 143 HL theorem, 47 hyperbolic angle-sum theorem, 154 hyperbolic cosine, 54 hyperbolic geometry, 154 hyperbolic parallel postulate, 154 hypotenuse, 36 is the longest side, 45 incenter, 137 incidence postulate, 4 incident with, 3 incircle, 137 for a triangle, 138 incircle theorem, 137 included angle, 35 included side, 35 induction, mathematical, 84 injective function, 5 inscribed circle, 137 for a triangle, 138 inscribed polygon, 136 interior of a circle, 129 of a convex polygon, 93 of a polygon, 87 of a ray, 16 of a segment, 11 of a simple region, 97 of an angle, 30, 34 interior angle adjacent, 43 alternate, 64 formed by a transveral, 64 of a triangle, 43 remote, 43 interior angle measure of a polygon, 92 interior diagonal, 93 intersecting lines, 3 intersection set, 167 inverse cosine, 51 inverse hyperbolic cosine, 54 inward-pointing ray, 90 isosceles right triangle, 122 isosceles triangle, 36 isosceles triangle theorem, 40 converse, 41 Pappus s proof, 41 Khayyam, Omar, 144, 153 Klein, Felix, 153 Lambert quadrilateral, 156, 157 leg of a right triangle, 36 of a Saccheri quadrilateral, 156 Legendre, Adrien-Marie, 144 Lemma B from the blog, 118 length

5 INDEX I-5 of a diameter, 130 of a segment, 11 of a vector, 51 lies on, 3 line, 2 contains infinitely many points, 7 in elliptic geometry, 56 in spherical geometry, 56 in the Poincaré disk model, 53 in the rational plane, 56 line geometry, 7 line segment, 11 line-circle theorem, 132 linear pair, 26 linear pair theorem, 26 converse, 34 linear triple, 29, 34 linear triple theorem, 29 Lobachevsky, Nikolai, 153 mathematical induction, 84 mean proportional, 125 measure of an angle, 2, 24 in the Poincaré disk model, 54 reflex, 86 standard, 86 meet, 3 midpoint, 14 midsegment, 157 neutral geometry, 49 noncollinear, 3 nonoverlapping regions, 97, 98 obtuse angle, 25 obtuse triangle, 36 octagon, 82 odd parity, 87 one-dimensional quantity, 114 one-point geometry, 56 one-to-one correspondence, 5 one-to-one function, 5 onto function, 5 open half-plane, 83 opposite angles, 74 opposite rays, 18 opposite sides of a quadrilateral, 74 opposite vertices of a quadrilateral, 74 outward-pointing ray, 90 overlapping regions, 97 Pappus of Alexandria, 40, 41 parallel lines, 3 are equidistant lines, 67 existence of, 65 parallel postulate Euclidean, 66, 140 Hilbert s, 140 hyperbolic, 154 parallel projection theorem, 113 parallel segments, 75 parallelism, transitivity of, 69, 140 parallelogram, 74 area of, 104 is convex, 77 properties, 79 parity of a ray, 87 Pasch s theorem, 36 pentagon, 82 perimeter of a polygon, 114 perimeter scaling theorem, 114 perpendicular bisector, 62 perpendicular bisector theorem, 62 perpendicular lines, 59 at a point, 59 constructing, 59 dropping, 59 plane separation postulate, 19 plane, the, 2 Playfair s postulate, 70 Poincaré disk, point, 2 in elliptic geometry, 56 in spherical geometry, 56 in the Poincaré disk model, 53

6 I-6 INDEX in the rational plane, 56 point of tangency, 131 polygon, 81 area of, 98 concave, 83 convex, 83, 84, 95 equiangular, 86 equilateral, 86 interior of, 93 regular, 86 polygon decomposition theorem, 99 polygon inequality, 134 polygonal region, 97 postulate angle measure, 24 area, 98, 105 distance, 4 Euclidean parallel, 66, 140 existence, 3 hyperbolic parallel, 154 incidence, 4 plane separation, 19 protractor, 25 ruler, 5 SAS, 39 set, 2 unit area, 98 Proclus s axiom, 140 Proclus s lemma, 68 projection of a leg, 125 proportion, 107 proportional, 107 protractor postulate, 25 Pythagorean postulate, 150 Pythagorean theorem, , , 150 converse, 122 quadrilateral, 73 concave, 75 convex, 75, 76 quadrilateral area scaling theorem, 115 radii, see radius radius, 129 ratio, 107 rational plane, 56 ray, 15 coordinate representation of, 16 lying in the interior of an angle, 30 lying on a side of a line, 21 rectangle, 74 area of, 102 properties, 79 rectangle decomposition lemma, 101 rectangular region, 97 reflex angle, 86 reflex measure of an angle, 86 reflexivity of congruence of angles, 28 of segments, 12 region determined by a polygon, 94, 97 polygonal, 97 simple, 97 regular polygon, 86 remote interior angle, 43 rhombus, 74 properties, 79 right angle, 25 right triangle, 36 area of, 103 isosceles, 122 right triangle proportion theorem, 125 right triangle similarity theorem, 124 ruler flipping lemma, 6 ruler placement theorem, 6 ruler postulate, 5 ruler sliding lemma, 5 SA congruence theorem for asymptotic triangles, 172 SAAAS, 80 SAASA, 80 Saccheri quadrilateral, Saccheri, Giovanni, 144, 177 Saccheri Legendre theorem, 144 for convex polygons, 145 SAS postulate, 39 SAS similarity theorem, 111 SAS theorem of Euclid, 38 SASAS, 78

7 INDEX I-7 SASSA, 80 scale factor, 108 scalene inequality, 44 scalene triangle, 36 scaling theorem diagonal, 114 height, 114 perimeter, 114 quadrilateral area, 115 triangle area, 115 secant line, 131, 133 segment, 11 contains infinitely many points, 15 coordinate representation of, 13 segment addition theorem, 12 segment construction theorem, 17 segment extension theorem, 17 segment subtraction theorem, 12 semiparallel segments, 75 set postulate, 2 side included between two angles, 35 of a line, 2, 19 in the Poincaré disk, 54 of a quadrilateral, 73 of a triangle, 35 of an angle, 23 opposite an angle, 35 side-angle-side postulate, 39 side-angle-side theorem, 38 side-side-side theorem, 38 side-splitter theorem, 108 converse, 111 similar polygons, 107, 108 similar triangle construction theorem, 111 similarity theorem AA, 110 right triangle, 124 SAS, 111 SSS, 111 simple region, 97 small angle lemma, 165 sphere, 56 square, 74 area of, 102 constructing, 119 diagonal of, 122 unit, 98 SSAAS, 79 SSASA, 80 SSS similarity theorem, 111 SSS theorem, 38, 41 SSS triangle construction theorem, 123 SSSAS, 79 standard measure of an angle, 86 starting point of a ray, 16 summit angles, 157 summit of a Saccheri quadrilateral, 157 supplementary angles, 26 surjective function, 5 symmetry of betweenness of points, 8 of betweenness of rays, 28 of equidistant lines, 68 tangent circles, 135 tangent line, 131 exterior to circle, 132 tangent line theorem, 131 tangent segment to a circle, 137 taxicab distance, 57 taxicab geometry, 57 Theorem A from the blog, 158 three-point plane, 56 transformation, 39 transitivity of congruence of angles, 28 of segments, 12 of triangles, 38 transitivity of parallelism, 69, 140 transversal, 64 trapezoid, 74 area of, 105 is convex, 76 triangle, 35 area of, 103 triangle area scaling theorem, 115

8 I-8 INDEX triangle dissection theorem, 105 triangle inequality, 45 triangle sliding lemma, 103 triangular region, 97 trichotomy for lines, 4 truncated triangle theorem, 77 two-circle theorem, 134 two-dimensional quantity, 115 ultraparallel lines, 167, 176 distance between, 176 ultraparallel theorem, 176 undefined terms, 2 uniqueness of common perpendiculars, 159 unit area postulate, 98 unit circle, 53 unit square, 98 vector, 50 displacement, 50 vertex concave, 86, 92 convex, 86, 92 of a quadrilateral, 73 of a triangle, 35 of an angle, 23 vertical angles, 27 vertical angles theorem, 27 vertices, see vertex Wallis s postulate, 141 Wallis, John, 141 weak angle-sum postulate, 150 whole greater than part angle, 28 converse, 16 segment, 12 X-theorem, 21 Y-theorem, 21