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30-60-90 Triangle, 70, 122 360 Theorem, 31 45-45-45 Triangle, 122 Index 30-60-90 triangle, 70, 122 opposite a side, 35 360 theorem, 31 right, 25 45-45-45 triangle, 122 angle addition theorem, 28 45-45-90 triangle, 70 angle bisector, 29, 63 60-60-60 triangle, 70 angle bisector proportionality theorem, 112 angle bisector theorem, 63 AA similarity theorem, 110 angle construction theorem, 25 AAA congruence theorem, 154 angle measure, 2, 24 AAASA, 80 between two lines, 175 AAS theorem, 46 in taxicab geometry, 57 AASAS, 78 in the Cartesian plane, 51 acute angle, 25 in the Poincar´edisk model, 54 acute triangle, 36 angle measure postulate, 24 additivity of defects, 146 angle subtraction theorem, 28 adjacent angles angle sum of a polygon, 82 for a convex quadrilateral, 78 of a quadrilateral, 74 for a polygon, 92 adjacent interior angle, 43 for a triangle, 69 adjacent sides angle-sum postulate, 142 of a polygon, 82 weak, 150 of a quadrilateral, 74 angle-sum theorem adjacent vertices for asymptotic triangles, 174 of a polygon, 82 for convex polygons, 84 of a quadrilateral, 74 for convex quadrilaterals, 78 admissible decomposition, 99 for general polygons, 93 all-or-nothing theorem, 151 for triangles, 69, 142 alternate interior angles, 64 hyperbolic, 154 alternate interior angles theorem, 64 arccosine, 51 converse, 67, 140 area, 2, 97–105 altitude of a triangle, 61, 103 of a parallelogram, 104 angle, 23 of a polygon, 98 acute, 25 of a polygonal region, 98 included between two sides, 35 of a rectangle, 102 obtuse, 25 of a right triangle, 103 of a polygon, 82 of a square, 102 of a quadrilateral, 73 of a trapezoid, 105 of a triangle, 35 of a triangle, 103 I-1 I-2 INDEX of a unit square, 98 Birkhoff, George D., 1 area postulate, 98 bisector independence of, 105 angle, 29, 63 unit, 98 perpendicular, 62 area scaling theorem Bolyai, J´anos, 153 quadrilateral, 115 boundary of a polygonal region, 97 triangle, 115 ASA theorem, 39 Cartesian plane, 50–53 ASASA, 79 center of a circle, 129 ASS non-theorem, 46, 48 characterizations of convex polygons, 95 asymptotic rays, 165 chord of a circle, 129 determine parallel lines, 166 chord theorem, 130 endpoint independence, 169 circle, 129 existence and uniqueness, 167 unit, 53 symmetry property, 168 circumcenter, 136 transitivity property, 170 circumcircle, 136 asymptotic triangle, 172 for a triangle, 137 angle-sum theorem, 174 circumcircle theorem, 136 exterior angle theorem theorem, 174 circumscribed circle, 136 SA congruence theorem, 172 for a triangle, 137 asymptotically parallel lines, 167 circumscribed polygon, 137 distance between, 177 Clairaut’s axiom, 148 existence and uniqueness, 175 Clairaut, Alexis, 148 closed half-plane, 83 base collinear points, 3 of a parallelogram, 104 collinear rays, 18 of a Saccheri quadrilateral, 156 common notions of a trapezoid, 104 for angles, 28 of a triangle, 103 for segments, 12 of an isosceles triangle, 36 of Euclid, 11 base angles, 36 common perpendicular, 65, 69, 160, 173, 176 Beltrami, Eugenio, 153 uniqueness, 159 betweenness common perpendiculars theorem, 65 consistency of, 9, 28 converse, 69 of numbers, 8 complementary angles, 26 of points, 8 components of a vector, 50 of rays, 27 concave polygon, 83 symmetry of, 8, 28 concave quadrilateral, 75 vs. betweenness, 33 concave vertex, 86, 92 vs. interior, 32 concentric circles, 129, 130 betweenness theorem concurrent, 136 converse, 9, 34, 45 congruent angles, 25 for points, 8 congruent polygons, 92 for rays, 28 congruent quadrilaterals, 74 bijective function, 5 congruent segments, 11 INDEX I-3 congruent simple regions, 97 diagonal scaling theorem, 114 congruent triangles, 37 diameter of a circle, 129, 130 consecutive interior angles, 64 length of, 130 consecutive interior angles theorem, 65 displacement vector, 50 converse, 67 distance, 2, 4 consistency from a point to a line, 63 of betweenness of points, 9 in taxicab geometry, 57 of betweenness of rays, 28 in the Cartesian plane, 51 constant of proportionality, 108 in the Poincar´edisk model, 54 constructing a perpendicular, 59 properties of, 7 constructing a square, 119 distance postulate, 4 contains (a point), 3 dodecagon, 82 convex polygon, 83, 84 dot product, 50 characterizations of, 95 dropping a perpendicular, 59 interior of, 93 convex quadrilateral, 75, 76 edge convex set, 20, 34 of a polygon, 82 convex vertex, 86, 92 of a quadrilateral, 73 coordinate elliptic geometry, 56 of a point, 5 endpoint of a ray, 25 of a ray, 16 coordinate function of a segment, 11 for a half-rotation, 25 equiangular polygon, 86 for a line, 5 equiangular triangle, 36, 41 coordinate representation equidistant, 159 of a ray, 16 from two lines, 63 of a segment, 13 from two points, 62 copying a quadrilateral, 78 equidistant lines, 67 copying a triangle, 41 are parallel, 67 corresponding angles, 64 symmetry of, 68 corresponding angles theorem, 65 equilateral polygon, 86 converse, 67 equilateral triangle, 36, 41 crossbar theorem, 37 equivalent postulates, 71, 139 cut by a transversal, 64 Euclid’s fifth postulate, 140 cyclic polygon, 136 implied by Euclidean parallel postulate, 70 implies Euclidean parallel postulate, 71 decagon, 82 Euclidean axioms, 139–151 defect of a polygon, 146 Euclidean geometry, 66 additivity of, 146 Euclidean parallel postulate, 66, 140 degrees, 24 implied by Euclid’s fifth postulate, 71 diagonal implies Euclid’s fifth postulate, 70 interior, 93 even parity, 87 of a polygon, 82 existence of a quadrilateral, 74 of infinitely many points, 7 of a square, 122 of parallels, 65 I-4 INDEX of three noncollinear points, 19 for a triangle, 138 of two distinct points, 3 incircle theorem, 137 existence postulate, 3 included angle, 35 exterior included side, 35 of a circle, 129 induction, mathematical, 84 of a polygon, 87 injective function, 5 exterior angle inscribed circle, 137 formed by a transversal, 64 for a triangle, 138 of a triangle, 43 inscribed polygon, 136 exterior angle theorem, 43 interior for asymptotic triangles, 174 of a circle, 129 external point for a line, 3 of a convex polygon, 93 of a polygon, 87 foot of a perpendicular, 61 of a ray, 16 four right angles theorem, 59 of a segment, 11 fourth angle of a Lambert quadrilateral, 157 of a simple region, 97 fourth vertex of a Lambert quadrilateral, 157 of an angle, 30, 34 interior angle Gauss, Carl Friedrich, 153 adjacent, 43 geometric mean, 125 alternate, 64 formed by a transveral, 64 half-plane, 20 of a triangle, 43 closed, 83 remote, 43 is convex, 20 interior angle measure of a polygon, 92 open, 83 interior diagonal, 93 half-rotation, 24 intersecting lines, 3 height intersection set, 167 of a parallelogram, 104 inverse cosine, 51 of a trapezoid, 104 inverse hyperbolic cosine, 54 of a triangle, 103 inward-pointing ray, 90 height scaling theorem, 114 isosceles right triangle, 122 hexagon, 82 isosceles triangle, 36 Hilbert’s parallel postulate, 140 isosceles triangle theorem, 40 hinge theorem, 143 converse, 41 HL theorem, 47 Pappus’s proof, 41 hyperbolic angle-sum theorem, 154 hyperbolic cosine, 54 Khayyam, Omar, 144, 153 hyperbolic geometry, 154 Klein, Felix, 153 hyperbolic parallel postulate, 154 hypotenuse, 36 Lambert quadrilateral, 156, 157 is the longest side, 45 leg of a right triangle, 36 incenter, 137 of a Saccheri quadrilateral, 156 incidence postulate, 4 Legendre, Adrien-Marie, 144 incident with, 3 Lemma B from the blog, 118 incircle, 137 length INDEX I-5 of a diameter, 130 onto function, 5 of a segment, 11 open half-plane, 83 of a vector, 51 opposite angles, 74 lies on, 3 opposite rays, 18 line, 2 opposite sides of a quadrilateral, 74 contains infinitely many points, 7 opposite vertices of a quadrilateral, 74 in elliptic geometry, 56 outward-pointing ray, 90 in spherical geometry, 56 overlapping regions, 97 in taxicab geometry, 57 in the Cartesian plane, 51 Pappus of Alexandria, 40, 41 in the Poincar´edisk model, 53 parallel lines, 3 in the rational plane, 56 are equidistant lines, 67 line geometry, 7 existence of, 65 line segment, 11 parallel postulate line-circle theorem, 132 Euclidean, 66, 140 linear pair, 26 Hilbert’s, 140 linear pair theorem, 26 hyperbolic, 154 converse, 34 parallel projection theorem, 113 linear triple, 29, 34 parallel segments, 75 linear triple theorem, 29 parallelism, transitivity of, 69, 140 Lobachevsky, Nikolai, 153 parallelogram, 74 area of, 104 mathematical induction, 84 is convex, 77 mean proportional, 125 properties, 79 measure of an angle, 2, 24 parity of a ray, 87 in taxicab geometry, 57 Pasch’s theorem, 36 in the Cartesian plane, 51 pentagon, 82 in the Poincar´edisk model, 54 perimeter of a polygon, 114 reflex, 86 perimeter scaling theorem, 114 standard, 86 perpendicular bisector, 62 meet, 3 perpendicular bisector theorem, 62 midpoint, 14 perpendicular lines, 59 midsegment, 157 at a point, 59 neutral geometry, 49 constructing, 59 noncollinear, 3 dropping, 59 nonoverlapping regions, 97, 98 plane separation postulate, 19 plane, the, 2 obtuse angle, 25 Playfair’s postulate, 70 obtuse triangle, 36 Poincar´edisk, 53–55 octagon, 82 point, 2 odd parity, 87 in elliptic geometry, 56 one-dimensional quantity, 114 in spherical geometry, 56 one-point geometry, 56 in taxicab geometry, 57 one-to-one correspondence, 5 in the Cartesian plane, 51 one-to-one function, 5 in the Poincar´edisk model, 53 I-6 INDEX in the rational plane, 56 ray, 15 point of tangency, 131 coordinate representation of, 16 polygon, 81 lying in the interior of an angle, 30 area of, 98 lying on a side of a line, 21 concave, 83 rectangle, 74 convex, 83, 84, 95 area of, 102 equiangular, 86 properties, 79 equilateral, 86 rectangle decomposition lemma, 101 interior of, 93 rectangular
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