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Abscissa, 203
Acute angles, 5
Acute triangle, 10
Addition postulate, 20
Addition method, 123
Adjacent angles, 12
Alternate interior angles, 49, 50–51
Alternation method, 123
Altitude:
to a side of a triangle, 10, 129, 133, 137, 170
of obtuse triangles, 11
Analytic geometry, 203–223
Angles, 4
acute, 5
adjacent, 12–13
alternate interior, 49–51
base, 10
bisectors, 6, 56–57, 83, 99, 126, 180, 195–196
bisector of a triangle, 10
central, 93, 95, 103–104, 106, 317
congruent, 6, 26, 50–52, 60, 68, 77–80, 104, 128–129, 256–7
constructing, 242
depression, of, 158
elevation, of 158
exterior, 48
inscribed, 104–106, 258–259, 317
measuring, 5
in a circle, 104–106
principles, 60
naming, 5
opposite, 105
plane, 266
reflex, 6
sum of measures in a triangle, 59–60, 256, 317
supplementary, 12–13, 26, 50–52, 62, 79, 105, 317
Angle-measure sum principles, 60–62
Angle-side-angle (ASA), 35, 246
Arcs, 4, 93, 95–96, 105, 226, 257, 317
length of, 185–186
major, 93
measuring, 103
minor, 93
Area, 164–171
of closed plane figures, 164
of equilateral triangle, 167, 318
of parallelogram, 165–166, 170, 263, 318
of quadrilateral, 214
of regular polygon, 182–183, 265, 318
of triangle, 166, 170, 214, 264, 318
Arms of a right triangle, 10
Assumptions, 234
Asymptote, slant, 305–308
Axis:
conjugate, 305–306
major, 300–301
minor, 300
of symmetry/reflection, 300, 302, 304–305
transverse, 305–307
Base angles:
of a trapezoid, 77–78
perpendicular, 83
Box (see Rectangular solid)
Center of circle, 3, 93, 96, 99, 179
Center of regular polygon, 179
Central angle, 4
of regular polygon, 179
Chords, 4, 93, 95–96, 105–106, 135, 226–227, 257, 259
Circles, 3, 93–120, 135–136, 186, 270–274, 297–298
angles, 104–106
central angle, 93, 95, 103–104, 106, 317
equal, 94
inequality theorems, 226–227
inscribed, 94
outside each other, 100
overlapping, 100
relationships, 93
sector of (see Sector)
segments (see Segments)
segments intersecting inside and outside, 135
small, 269
tangent externally, 100
tangent internally, 100
Circumference of circle, 3, 93, 184
Circumscribed polygon, 94, 179
Combination figure, area of, 188
Complementary angles, 12–13, 26, 317
Conclusion, 27
volume, 276
Conic section, 297–311
degenerate, 297–298
reflection properties, 299, 302, 304–305, 308–309
Congruent angles, 6, 26, 50–52, 60, 68, 77–80, 104, 128–129, 256–257
Congruent circles, 4, 94–96, 104
Congruent figures, 34
Congruent polygons, 169
Congruent segments, 3, 9, 39, 77–80, 84–85, 99
Congruent triangles, 34–47, 83
principles, 34–35
methods of proving, 35
selecting corresponding parts, 34
Construction, 241–254
angle, 242
bisectors, 243–244
center of a circle, 249
circumscribed circle, 249
inscribed circle, 250
inscribed polygon, 251–252
line segment congruent to a given linesegment, 242
parallel lines, 248
perpendiculars, 243–244
similar triangles, 252
tangents, 249
triangles, 245–246
Continued ratio, 121
Contrapositive of statement, 235
partial, of theorem, 237
Coordinates, 203
Corollary, 39
Corresponding:
angles, 128
Cosine (cos), 154
Cube, 266–267
surface area, 274
volume, 276
Curvature, 315
surface area, 275
volume, 276
Decagon, 65–66
Deduction, 18
Deductive reasoning, 18
Definitions, 233
requirements, 233
Diagonals, 79–80, 82–83, 142, 319
congruent, 83
Dihilations, 292
Distances, 271
between two geometric figures, 55
between two points, 55
principles of, 55–57
Division postulate, 21
Dodecagon, 65
Dual statements, 270–274
Eccentricity, 298
Edge, 266
Elements, The, 312
Ellipse, 297–302
Elliptic geometry, 314–315
Enlargement, 292
Equal polygons, 169
Equation of degree two in variables x and y, 298
Equator, 269
Equilateral:
parallelogram, 82
triangles, 9–10, 39–40, 140–141, 181, 246, 318
angle measure, 61
Exterior:
angles, 48
sides, 13
Extreme, 122–123
Face, 266
lateral, 267
Fifth postulate problem, 313–314
Focus, 298–308
Fourth proportionals, 122
Frustum, 268
Graph, 203
Heptagon, 65
History of geometry, 1
Horizontal line, 158
Hyperbolic geometry, 315–316
Hypotenuse, 10, 86, 137–138, 141
Hypotenuse-leg (hy-leg), 68, 247, 257
Hypothesis, 27
Identity postulate, 20
If-then form, 27
Image, 281
Inclination of a line (see Slope)
Indirect reasoning, 229
Inequality, 224
axioms, 224–227
postulate, 225
Inscribed angle, 104–106, 258–259, 317
Inscribed circle, 94
Inscribed polygon, 94
Intercept an arc, 93
Inverse:
partial, of theorem, 237
of statement, 235
Inversion method, 123
Isosceles right triangle, 61, 141–142
Isosceles trapezoid, 77–78
Isosceles triangles, 9, 39, 141, 256
Latitude, 269
Legs of a right triangle, 10, 137–138, 141
Length formulas, 21
centers of two circles, of, 100
naming, 1
parallel (see Parallel lines)
sight, of, 158
Line segments, 56
naming, 2
Line symmetry, 286
in analytic geometry, 212–213
fundamental theorems, 195–196
Logically equivalent statements, 235–236
Longitude, 269
Mean of a proportion, 122
in a right triangle, 137
Measurement (see specific applications)
of segment, 22
Minor arc, 93
Minutes, 5
Multiplication postulate, 21
Necessary conditions, 238
Negative curvature, 315
Negative of a statement, 235
N-gon, 64
sum of angles, 317
Nonagon, 65
Non-euclidean geometries, 312–316
Number line, 203
Obtuse triangle, 10
Octagon, 65
Ordinate, 203
Origin, 203
Pairs of angles, principles, 13
Parallel lines, 48, 79, 85–86, 105, 125–126, 170, 196
constructing, 248
postulate, 49
principles of, 49–52
three or more, 51–52
Parallelepiped, 267
Parallelograms, 79–80, 82–84, 170
area of, 165–166, 170, 263, 318
sides, 79
Partial converse of theorem, 237
Partial inverse of theorem, 237
Partition postulate, 20
Perpendicular, 6, 22, 50–51, 62, 99, 129, 226, 270–274
constructions, 243–244
Perpendicular bisector, 6, 55–56, 96, 195
of a side of a triangle, 10
Plane, 270–274
Points, 1
angle principles, 66–67
central angle, of, 180
congruent, 169
equal, 169
inscribed, 94
interior and exterior angles, 60, 65–66, 180
regular (see Regular polygons)
sum of measures:
of exterior angles, 66
of interior angles, 65
Polyhedron, 266
regular, 269–270
Positive curvature, 315
algebraic, 20
geometric, 21
Powers postulate, 21
Principle, 25
Prism, 267
right, 267
volume, 276
Projection, 137
Projective space, 315
Proof, 18–19
by deductive reasoning, 18
Proportional:
fourth, 122
segments, 125
principles, 125–126
changing into new proportions, 123
eight arrangements, 125
extreme, 122–123
mean, 122–123
similar triangles, 128–129
Pyramid, 266–267
regular, 268
volume, 276
Pythagorean Theorem, 138, 262–263, 318
Quadrants, 203
Quadrilaterals, 9, 65, 79, 80, 105
area of, 214
Radius:
of regular polygon, 179
Ratios, 121
continued, 121
of segments and area of regular polygons, 183
of similitude, 133
Ray 1
Reasoning:
deductive, 18
indirect, 229
Rectangle, 82–83
Rectangular solid, 267
surface area, 275
volume, 276
Reflection, 284
properties, 299, 302, 304–305, 308–309
Reflectional symmetry, 286
Reflex angle, 6
Reflexive postulate, 20
Regular polygons, 65–66, 179–183, 270
center of, 179
central angle of, 179
inscribed, 94
principles, 179–180
radius of, 179
Rhombus, 82–83
Right triangles, 10, 61, 68, 82, 86, 105, 129, 138, 154, 318
mean proportionals in, 137, 262
principles of, 30°-60°-90° triangle 140–141
principles of, 45°-45°-90° triangle 61, 141–142
Rigid motion, 290
Rotational symmetry, 289
Scale factor, 292
Scalene triangle, 9
Scaling, 292
Secants, 93, 105–106, 136, 260, 317–318
Seconds, 5
Section of a polyhedron, 266
Sector, 185
Segments, 186
external, 136
proportional, 125
Side-angle-angle (SAA), 68, 247
Side-angle-side (SAS), 35, 246
Similar polygons, 128, 133, 169, 171
Similar triangles, 128–134, 261
constructing, 252
principles, 128–129
proportion, 129
Sine (sin), 154
Slope:
of parallel and perpendicular lines, 210
principles, 209–211
Small circle, 269
Solid, 266
geometry principles, 270
equation, 274
surface area, 275
volume, 276
Square unit, 164
contrapositive of, 235
dual, 270–274
inverse of, 235
logically equivalent, 235–236
negative of, 235
Substitution postulate, 20
Subtraction method, 123
Subtraction postulate, 21
Sufficient conditions, 238
Sum of measures of angles:
exterior angles of polygons, 66
interior angles of polygons, 65
triangle, of a, 59–60, 256, 317
Supplementary angles, 12–13, 26, 50–52, 62, 79, 105, 317
Syllogistic reasoning, 18
Symmetry, axis of, 284
Tangent:
trigonometric function (tan), 154, 209
to a circle, 93, 99, 105–106, 136, 260–261, 317–318
principles, 99
partial converses of, 237
partial inverses of, 237
combination of, 290
image of, 281
notation, 281
Transitive postulate, 20
Translation, 282
isosceles, 77–78
Triangles, 9, 56–57, 61, 65, 85–86, 125–126
acute, 138
altitude, 10, 129, 133, 137, 170
area of, 166, 170, 214, 264, 318
equilateral (see Equilateral triangles)
exterior angles, 61
inequality theorems, 225–226
perpendicular bisector of side, 56
right (see Right triangles)
scalene, 9
similar (see Similar triangles)
sum of angles, 59–60, 256, 317
Trigonometry, 154–163
table of values, 320
Truth, determining, 28
Vertical angles, 12–13
Volume, 276
x-axis, 203
x-coordinate, 203
y-axis, 203
y-coordinate, 203
y-intercept, 212
Zero curvature, 315