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D is a point on side and E is a point on side. If AD 6, DB 9, and AC 20, find AE and EC. AB AC DE BC of ABC such that 12. A pile of gravel is in the form of a cone. The circumference of the pile of gravel is 75 feet and its height is 12 feet. How many cubic feet of gravel does the pile contain? Give your answer to the nearest hundred cubic feet. Part III Answer all questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 13. a. On graph paper, sketch the graph of y x2 x 2. b. On the same set of axes, sketch the graph of y x 5. c. What are the common solutions of the equations y x2 x 2 and y x 5? 14. a. Show that the line whose equation is x y 4 intersects the circle whose equation is x2 y2 8 in exactly one point and is therefore tangent to the circle. b. Show that the radius to the point of tangency is perpendicular to the tangent. 14365C14.pgs 7/10/07 10:00 AM Page 635 Cumulative Review 635 Part IV Answer all questions in this part. Each correct answer will receive 6 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 15. a. The figure below shows the construction of the perpendicular bisector, g DE, of segment AB. Identify all congruent segments and angles in the g DE is the perpen- construction and state the theorems that prove that dicular bisector of. AB E A M D B b. The figure below shows the construction of FDE congruent to CAB. Identify all congruent lines and angles in the construction and state the theorems that prove that FDE CAB. C B A F E D 16. a. Quadrilateral ABCD is inscribed in circle O, AB CD, and AC is a diameter. Prove that ABCD is a rectangle. b. In quadrilateral ABCD, diagonals AC and CDE are similar but not congru |
ent, prove that the quadrilateral is a trapezoid. intersect at E. If ABE and BD c. Triangle ABC is equilateral. From D, the midpoint of AB, and to F, the midpoint of, a line BC segment is drawn to E, the midpoint of AC. Prove that DECF is a rhombus. 14365Index.pgs 7/13/07 10:19 AM Page 636 INDEX A AA triangle similarity (AA), 490. See also Angle-angle triangle similarity AAS triangle congruence, 352. See also Angle-angle-side triangle congruence Abscissa, 210 Absolute value, 7–8 Acute angle, 17 Acute triangle, 25, 105 Addition of angles, 20–21 associative property of, 4–5 closure property of, 4 commutative property of, 4 distributive property of multiplication over, 5 inequality postulates involving, 267–268 of line segments, 12–13 Addition postulate, 119–120, 267–268 Additive identity, 5 Additive inverses, 5 Adjacent angles, 145 Adjacent interior angle, 277 Adjacent sides, 380 Adjacent vertices, 368, 380 Algebra, Boolean, 34 Alternate exterior angles, 330 Alternate interior angles, 330 Altitude(s). See also Height of cone, 456 of cylinder, 453 of pyramid, 449 of rectangle, 389 of triangle, 175 concurrence of, 317–320 Analytic geometry, 209 Angle(s), 15 acute, 17, 105 addition of, 20–21 adjacent, 145 alternate exterior, 330 alternate interior, 330 base, 25 bisectors of, 20, 103, 137, 199 central, 536–537 classifying according to their measures, 17 classifying triangles according to, 25–26 complementary, 145 congruent, 19, 103 construction of, congruent to a given angle, 197 corresponding, 330 defined, 15 definitions involving pairs of, 145–146 dihedral, 424 exterior, of, 16, 330 polygon, 276–277, 369 triangle, 277–279 formed by tangent and chord, 567–568 formed by tangent and secant, 568–571 formed by two intersecting chords, 568 inequalities involving, for triangle, 281–284 inscribed, and their measures, 552–555 interior, of, 16, 330 polygon, 368–369 linear pair of, 148 measure of |
, 16 naming of, 16 obtuse, 17 plane, 424 postulates of, 135–138 proving theorems about, 144–145 right, 17 sides of, 15 straight, 16, 17 subtraction of, 20–21 sum of measures of, of triangle, 347–350 supplementary, 145–146, 148–149 theorems involving pairs of, 146–148 trisection of, 604 using congruent triangles to prove congruent, 178–179 vertex of, 15, 25 vertical, 149 Angle addition postulate, 120 Angle-angle-side triangle congruence, 352. See also AAS triangle congruence Angle-angle triangle similarity, 490. See also AA triangle similarity Angle bisector(s) concurrence of, of triangle, 364–365 of triangle, 176 Angle measure, preservation of under dilation, 497 under glide reflection, 244 under line reflection, 217 under point reflection, 228 under rotation about a fixed point, 239 under translation, 234 Angle-side-angle triangle congruence, 162. See also ASA triangle congruence Antecedent, 55 Apothem of circle, 547 of polygon, 567 Arc(s) of circle, 537 congruent, 538 degree measure of, 537–538 intercepted, 537 major, 537 minor, 537 types of, 537 Arc addition postulate, 539 Archimedes, 535 Area of a polygon, 409–410 Argument, valid, 75, 109 Arrowheads, 2 ASA triangle congruence, 162. See also Angle-side-angle triangle congruence Associative property of addition, 4–5 of multiplication, 5 Axiom, 93, 109–110 Axis of symmetry, 219, 625 B Base(s) of cone, 456 of cylinder, 453 of polyhedron, 440 of rectangle, 389 of regular pyramid, 450 of trapezoid, 402 Base angles of isosceles trapezoid, 403 of isosceles triangle, 25 lower, 404 upper, 404 Basic constructions, 196–202 Baudha¯ yana, 474 Betweenness, 8 14365Index.pgs 7/13/07 10:19 AM Page 637 Biconditional(s), 69–73 applications of, 70–73 definitions as, 97–99 Bisector of angle, 20, 103, 137 of line segment, 12 Bolyai, János, |
379 Boole, George, 34 Boolean algebra, 34 C Cartesian coordinates, 209 Cavalieri, Bonaventura, 419 Cavalieri’s Principle, 419 Center of circle, 460, 536 chords equidistant from, 546–550 of regular polygon, 450 of sphere, 459 Center-radius equation of a circle, 582 Centimeter, cubic, 446 Central angle of circle, 536–537 Centroid, 506 Chord(s) angles formed by, 567–568 of circle, 543 equidistant from center of circle, 546–550 segments formed by two intersecting, 575 Circle(s), 460 apothem of, 547 arc of, 537 center of, 460, 536 central angle of, 536–537 chords of, 543, 546–550 circumference of, 535 concentric, 612 congruent, 538 in coordinate plane, 581–584 defined, 536 diameter of, 543 exterior of, 536 inscribed angles of, 552–553 interior of, 536 polygons circumscribed about, 563–564 polygons inscribed in, 550–551 radius of, 536 secant of, 558 squaring of, 604 tangent to, 558 Circular cylinder, 453 surface area of, 454 volume of, 454 Circumcenter, 194 Circumference of circle, 535 Circumscribed circle, 550 Circumscribed polygon, 563 Closed sentence, 37 Closure property of addition, 4 of multiplication, 4 Collinear set of points, 7 Collinearity, preservation of under line reflection, 217 under translation, 234 Common external tangent, 560 Common internal tangent, 559 Common tangent, 559–561 Commutative property of addition, 4 of multiplication, 4 Compass, 196, 604 Complementary angles, 145 Complements, 145 Composition of transformations, 243, 251–252 Compound locus, 614 Compound sentence(s), 42, 53 conjunctions as, 42–46 Compound statement(s), 42, 53 conditionals as, 4253–57 disjunctions as, 48–50 Concave polygon, 368 Concentric circles, 612 Conclusion(s), 55 drawing, 80–83 Concurrence, 193 of altitudes of triangle, 317–320 of angle bisectors of triangle, 364–365 Conditional(s), 53 |
–57, 60, 335 contrapositive of, 64–65 converse of, 63–64 false, 63, 65 hidden, 55–57, 98 inverse of, 61–62 parts of, 55 as they relate to proof, 138–139 true, 63, 64 truth value of, 54–55 Cone(s), 456–458 altitude of, 456 base of, 456 frustum of, 459 height of, 456 right circular, 456 slant height of, 456 surface area of, 457–458 volume of, 457–458 Congruence equivalence relations of, 156–157 of triangles, 134, 174–203 Congruent angles, 19, 103 Congruent arcs, 538 Congruent circles, 538 Congruent polygons, 154–155 corresponding parts of, 155 Congruent segments, 9 Congruent triangles, 155–156 in proving line segments congruent and angles congruent, 178–179 using two pairs of, 186 Conjecture, 95, 97 Conjunct, 42 Conjunction, 42–46 Consecutive angles of polygon, 368 of quadrilateral, 380 Consecutive sides of quadrilateral, 380 Consecutive vertices of polygon, 368 of quadrilateral, 380 Consequent, 55 Constant of dilation, k, 501 Constant of proportionality, 486 Index 637 Constructions of angle bisector, 199 of congruent angles, 197 of congruent segments, 196–197 locus in, 609–611 of midpoint, 198 of parallel lines, 605–606 of perpendicular bisector, 198 of perpendicular through a point not on the line, 201 of perpendicular through a point on the line, 200 Contraction, 496 Contrapositive, 64–65 Converse, 63–64 of Isosceles Triangle Theorem, 357–360 of Pythagorean Theorem, 516–517 Converse statement, 335 Convex polygon, 368 Coordinate geometry, 209, 290 equidistant lines in, 619–622 points at fixed distance in, 616–618 Coordinate of point, 4, 7 Coordinate plane, 210 circles in, 581–584 dilations in, 247–248 line reflections in, 222–225 locating points in, 210–211 parallel lines in, 342–344 point reflections in, 227–231 rotations in, 238–242 secants in, 588– |
591 tangents in, 588 translations in, 232–235 preservation of angle measure, 232 preservation of collinearity, 232 preservation of distance, 232 preservation of midpoint, 232 Coordinate proof, 313–314 of general theorems, 313 for special cases, 313 Coordinates, 210 of point in plane, 210–213 rectangular, 211 Coplanar lines, 329 Corollary, 182 Corresponding angles, 155, 330 Corresponding parts of congruent polygons, 155 Corresponding sides, 155 Counterexample, 95 Cube, 445 duplication of, 604 Cubic centimeter, 446 Cylinder, 453–455 altitude of, 453 bases of, 453 circular, 453 height of, 453 lateral surface of, 453 right circular, 453 D Decagon, 367 Deductive reasoning, 97, 100–103, 150–151 Definition(s), 7, 93 as biconditionals, 97–99 qualities of good, 7 14365Index.pgs 7/13/07 10:19 AM Page 638 638 Index Definition(s) cont. using, in proofs, 141–142 writing as conditionals, 98 Degree measure of angle, 16 of arc, 537–538 DeMorgan, Augustus, 34 DeMorgan’s Laws, 34 Descartes, René, 209, 290 Detachment, law of, 75, 101, 105 Diagonal of polygon, 368 of quadrilateral, 380 Diagram(s) tree, 42 using, in geometry, 26–27 Diameter, 543 Dihedral angle, 424 Dilation(s), 495–499 in coordinate plane, 247–248 defined, 247 preservation of angle measure under, 496–497 preservation of collinearity under, 498 preservation of midpoint under, 497 Direct isometry, 252–253 Direct proof, 105–106 Disjunct, 48 Disjunction, 48–50, 76 Disjunctive inference, law of, 76–78 Distance, between two parallel lines, 383 between two planes, 437 between two points, 7–8 from a point to a line, 20 preservation of, under glide reflection, 244 preservation of, under point reflection, 228 preservation of, under rotation about a fixed point, 239 preservation of, under translation, 234 Distance formula, 521–522, 522 Distance postulate, 136 Distributive property, 5 Divide and average method, 174 Division postulate, |
124, 270–271 Dk, 247, 496 Domain, 36, 250 E Edge of polyhedron, 440 Elements (Euclid), 1, 262, 474 Endpoint, of ray, 15 Enlargement, 496 Epicureans, 262 Equality reflexive property of, 110–111 symmetric property of, 111 transitive property of, 111, 263–264 Equation(s) of line, 295–299 solving, with biconditionals, 70–71 Equiangular triangle, 25 Equidistant, 191 Equidistant lines in coordinate geometry, 619–622 Equilateral triangle(s), 24, 25, 181–183 properties of, 183 Equivalence relation, 111 of congruence, 156–157 of similarity, 487–488 Eratosthenes, 1 Euclid, 1, 93, 134, 262, 328, 379, 535 parallel postulate of, 328 Euclid Freed of Every Flaw, 379 Exclusive or, 51 Exterior angle(s), 330 alternate, 330 of polygon, 276–277, 369–371 of triangle, 277–279 Exterior angle inequality theorem, 277 Exterior angle theorem, 349 Exterior of angle, 16 Exterior of circle, 536 External segment, 576 Extremes, 476 F Face(s) of polyhedron, 440 Fermat, Pierre de, 209 Fixed point, 214 Foot of altitude, 525 Foot of perpendicular, 20 Formula(s) angles, central, 555 inscribed, 555 formed by tangents, chords, and secants, 571–572 of polygons, 369 area of a rectangle, 409 circle, 582 distance, 522 Heron’s, 174 lateral area, of cone, 457, 467 of cylinder, 454, 467 of prism, 442, 467 of pyramid, 467 midpoint, 304 point-slope, 297 segments formed by tangents, chords, and secants, 579 slope, 292, 297 surface area, of cone, 457 of cylinder, 454, 467 of prism, 442, 467 of pyramid, 467 of sphere, 462, 467 volume, of cone, 457, 467 of cylinder, 454, 467 of prism, 446, 467 of pyramid, 449, 467 of sphere, 462, 467 Foundations of Geometry (Hilbert), 93 45-45-degree right triangle |
, 517–518 Frustum of cone, 459 Function(s) defined, 250 transformations as, 250–254 G Galileo, 419 Generalization, 94 General quadrilateral, 380 Geometric constructions, 196. See also Constuctions Geometric inequalities, 262–285 basic inequality postulates, 263–265 inequalities involving lengths of the sides of a triangle, 273–274 inequalities involving sides and angles of a triangle, 281–284 inequality involving an exterior angle of a triangle, 276–279 inequality postulates involving addition and subtraction, 267–268 inequality postulates involving multiplication and division, 270–271 Geometric mean, 478 Geometry analytic, 209 coordinate, 209, 290 deductive reasoning, 100–103 defined, 2 definitions as biconditionals, 97–99 inductive reasoning, 94–97 non-Euclidean, 376 spherical, 32 proving statements in, 93–130 addition and subtraction postulates, 118–122 direct proofs, 105–108 indirect proofs, 105–108 multiplication and division postulates, 124–126 postulates, theorems, and proof, 109–115 substitution postulate, 115–117 solid, 420 using diagrams in, 26–27 using logic to form proof, 100–103 Glide reflection, 243–245 Graphing polygons, 212–213 Graphs, 4 Great circle of sphere, 460, 461 H Half-line, 14–15 Heath, Thomas L., 1 Height. See also Altitude of cone, 456 of cylinder, 453 of prism, 440 of pyramid, 449 Heron of Alexandria, 174 Heron’s formula, 174 Hexagon, 367 Hidden conditional, 55–57, 98 Hilbert, David, 93 HL triangle congruence theorem, 362–365 Hypotenuse, 26 Hypotenuse-leg triangle congruence theorem, 362–365. See also HL triangle congruence theorem Hypothesis, 55 I Identity additive, 5 14365Index.pgs 7/13/07 10:19 AM Page 639 multiplicative, 5 Identity property, 5 If p then q, 53 Image, 214, 215 Incenter, 364 Included angle, 24 Included side, 24 Inclusive or, 51 Incomplete sentences, 35 Indirect proof, 105–108, 283, 309, 331, 336, 425, 429, 431, 434, 436, 559 Inductive reasoning, 94–97 Inequality geometric, 262– |
5 involving exterior angle of triangle, 276–279 involving lengths of sides of triangle, 273–274 Lateral sides of prism, 440 Lateral surface of cylinder, 453 Law(s) DeMorgan’s, 34 of Detachment, 75, 101, 105 of Disjunctive Inference, 76 of logic, 35, 74–78 Leg(s) proving right triangles congruent by hypotenuse, 362–365 of right triangle, 26 of trapezoid, 402 Leibniz, Gottfried, 34, 290 Length of line segment, 9 Line(s), 1, 2, 7, 420–422 coplanar, 329 equation of, 295–299 equidistant, in coordinate geometry, involving sides and angles of triangle, 281–284 transitive property of, 264 Inequality postulate(s), 263, 265 619–622 number, 3–4 order of points on, 8 parallel, 421, 605–606 involving addition and subtraction, in coordinate plate, 342–344 267–268 involving multiplication and division, 270–271 relating whole quantity and its parts, 263 parallel to a plane, 433 perpendicular, 20, 100, 149 methods of proving, 193 planes and, 423–431 slopes of, 307–310 transitive property, 263–264 points equidistant from point and, Inscribed angle of circle, 552–553 measures of, 552–555 Inscribed circle, 563 Inscribed polygon, 550 Intercepted arc, 537 Interior angle(s), 330 alternate, 330 on the same side of the transversal, 330 of polygon, 368–369 Interior of angle, 16 Interior of circle, 536 Intersecting lines, equidistant from two, 621–622 624–629 postulates of, 135–138 skew, 421–422 slope of, 291–294 straight, 1, 2 Linear pair of angles, 148 Line of reflection, 214 Line reflection(s), 214–220 in coordinate plane, 222–225 preservation of angle measure under, 217 preservation of collinearity under, 217 preservation of distance under, 215 preservation of midpoint under, 217 Intersection of perpendicular bisectors Line segment(s), 9 of sides of triangle, 193–195 Inverse of a conditional, 61 Inverse property, 5 Inverses, 61–62 additive, 5 multiplicative, 5 Isometry, 244 direct, 252–253 opposite, 253 Isosceles |
quadrilateral, 379 Isosceles trapezoid(s) base angles of, 403, 404 properties of, 403 proving that quadrilateral is, 403–407 Isosceles triangle(s), 24, 25, 181–183, 451 base angle of, 25 parts of, 25 vertex angle of, 25 Isosceles triangle theorem, 181 converse of, 357–360 L Lateral area of prism, 442 Lateral edges of prism, 440, 441 addition of, 12–13 associated with triangles, 175–177 bisector of, 12 congruent, 9 construction of congruent segment, 196–197 construction of perpendicular bisector and midpoint, 198 divided proportionally, 482 formed by intersecting secants, 576–578 formed by tangent intersecting secant, 575–576 formed by two intersecting chords, 575 length or measure of, 9 on a line, projection of, 510 methods of proving perpendicular, 193 midpoint of, 11–12, 300–305 perpendicular bisector of, 191–195 postulates of, 135–138 proportions involving, 480–484 subtraction of, 12–13 tangent, 561–563 Index 639 using congruent triangles to prove congruent, 178–179 Line symmetry, 218–220 Lobachevsky, Nicolai, 379 Locus, 604–630 compound, 614 discovering, 610–611 equidistant from two intersecting lines, 613 equidistant from two parallel lines, 614 equidistant from two points, 613 fixed distance from line, 614 fixed distance from point, 614 meaning of, 609–611 Logic, 34–92 biconditionals in, 69–73 conditionals in, 53–57 conjunctions in, 42–46 contrapositive in, 64–65 converse in, 63–64 defined, 35 disjunctions in, 48–50 drawing conclusions in, 80–83 equivalents in, 65–67 in forming geometry proof, 100–103 inverse in, 61–62 law(s) of, 35, 74–78 detachment, 75 disjunctive inference, 76–78 negations in, 38 nonmathematical sentences and phrases in, 35–36 open sentences in, 36–37 sentences and their truth values in, 35 statements and symbols in, 37 symbols in, 38–39 two uses of the word or, 51 Logical equivalents, 65– |
67 Lower base angles of isosceles trapezoid, 404 M Major arc, 537 Mathematical sentences, 35 Mean(s), 476 geometric, 478 Mean proportional, 478–479, 512 Measure of angle, 16, 17, 552–555 formed by tangent intersecting a secant, 568 formed by two intersecting chords, 568 formed by two intersecting secants, 569 formed by two intersecting tangents, 569 of arc, 537 of central angle of a circle, 537 of inscribed angle of a circle, 553 of segments formed by intersecting chords, 575 intersecting secants, 576–577 tangent intersecting a secant, 575–576 Median of trapezoid, 405 14365Index.pgs 7/13/07 10:19 AM Page 640 640 Index Median cont. of triangle, 175–176 Metrica (Heron of Alexandria), 174 Midpoint of line segment, 11–12, 300–305 preservation of under dilation, 497 under glide reflection, 244 under line reflection, 217 under point reflection, 228 under rotation about a fixed point, 239 under translation, 234 Midpoint formula, 304 Midsegment, 346–347, 480 Midsegment theorem, 480 Minor arc, 537 Multiplication associative property of, 5 closure property of, 4 commutative property of, 4 distributive property of, 5 inequality postulates involving, 270–271 Multiplication postulate, 124, 270–271 Multiplication property of zero, 5 Multiplicative identity, 5 Multiplicative inverses, 5 N Negation, 38 Negative rotation, 239 Negative slope, 293 Newton, Isaac, 290 n-gon, 367 Noncollinear set of points, 7 Nonadjacent interior angle, 277 Nonmathematical sentences, phrases and, 35–36 No slope, 293 Number(s) properties of system, 110 rational, 3 real, 3 Number line, 3–4 distance between two points on the real, 8 Numerical operation, 4 O Obtuse angle, 17 Obtuse triangle, 25 Octagon, 367 One-to-one correspondence, 154 Open sentence, 36–37 Opposite angle, 24 Opposite isometry, 253 Opposite rays, 15 Opposite side, 24 Or exclusive, 51 inclusive, 51 Order of points on a line, 8–9 Ordered pair, 210 Ordinate, 210 Orientation, 252–254 |
Origin, 210 Orthocenter, 319 P Parabola, turning point of, 626 Paragraph proof, 101–102 Parallelepiped(s), 441, 442 rectangular, 442 volume of, 446 Parallel lines, 328–372, 421 constructing, 605–606 in coordinate plane, 342–344 defined, 329 equidistant from two, 621 methods of proving, 333 planes and, 433–439 properties of, 335–340 proving, 329–333 in space, 421 transversals and, 330–332 Parallelogram(s), 380–383 proof of quadrilateral as, 385–387 properties of, 383 Parallel planes, 438 Partition postulate, 118–119 Pentagon, 23, 367 Perpendicular bisector construction of, 198 of line segment, 191–195 Perpendicular Bisector Concurrence Theorem, 193 Perpendicular lines, 20, 100, 149 construction of through given point on line, 200 through point not on the given line, 201 methods of proving, 193 planes and, 423–431 slopes of, 307–310 Phrase(s), 35 nonmathematical sentences and, 35–36 Plane(s), 420–422. See also Coordinate plane coordinates of point in, 210–213 defined, 2 distance between two, 437 naming of, 2 parallel, 438 symmetry, 464 Plane angle, 424 Playfair, John, 328 Playfair’s postulate, 328 Point(s), 1, 2, 7, 420–422 collinear, 7 coordinate of, 4, 7 distance between two, 7–8 equidistant from two, 619–620 finding coordinates of, in plane, 211 fixed, 214 at fixed distance in coordinate geometry, 616–618 locating, in coordinate plane, 210–211 noncollinear set of, 7 order of, on a line, 8 points equidistant from line and, 624–629 projection of, on line, 510 Point reflection, 227–229 in coordinate plane, 229–231 properties of, 228–229 Point reflections in the coordinate plane, 247–248 preservation of angle measure under, 228 preservation of collinearity under, 228 preservation of distance under, 228 preservation of midpoint under, 228 Points equidistant from point and line, 624–629 Point-slope form of equation of line, 297 Point symmetry, 229 Polygon(s), 23, 367 areas of, 409– |
410 circumscribed about a circle, 563–564 concave, 368 convex, 368 diagonals of, 368 exterior angles of, 276–277, 369–371 graphing, 212–213 inscribed in circle, 550–551 interior angles of, 368–369 regular, 369 sides of, 23 similar, 486–488 Polyhedron(s), 440–442 bases of, 440 edges of, 440 faces of, 440 vertices of, 440 Positive rotation, 239 Positive slope, 292 Postulate(s), 93, 109–110 addition, 119–120, 267–268, 539 basic inequality, 263–265 of the coordinate plane, 210 division, 124, 270–271 Euclid’s parallel, 328 first, in proving statements, 110–113 of lines, line segments, and angles, 135–138 multiplication, 124, 270–271 partition, 118–119 Playfair’s, 328 powers, 124 relating a whole quantity and its parts, 263 roots, 125–126 of similarity, 490 substitution, 115–117, 264 subtraction, 121–122, 267–268 trichotomy, 264–265 using, in proofs, 141–142 Powers postulate, 124 Preimage, 214, 215 Premise, 55, 75 Prism, 440 altitude of, 440 height of, 440 lateral area of, 442 lateral edges of, 440, 441 lateral sides of, 440 right, 441 surface area of, 440–443 total surface area of, 442 volume of, 446–447 14365Index.pgs 7/13/07 10:19 AM Page 641 Projection of point on a line, 510 of segment on a line, 510 Proof(s), 100 conditional statements as they relate to, 138–139 by contradiction, 105 direct, 105–106 indirect, 106–108, 283, 309, 331, 336, 425, 429, 431, 434, 436, 559 paragraph, 101–102 transformational, 308 two-column, 101 using postulates and definitions in, 141–142 Proportion(s), 476–477 extremes of, 476 involving line segments, 480–484 in right triangle, 510–513 Proportionality, constant of, 486 Proving triangles congruent by AAS, 352–355 ASA, 161–163 SAS, 158–159 SSS, 165 HL, 362–365 similar by, 489–494 AA, 489–494 |
SSS, 489–494 SAS, 489–494 Pyramid(s), 449 altitude of, 449 height of, 449 properties of regular, 450–451 regular, 449 slant height of, 449 surface area of, 449 vertex of, 449 volume of, 449 Pythagoras, 1, 474 Pythagorean Theorem, 474, 515 converse of, 516–517 Pythagorean triple, 517 Q Quadrilateral(s), 23, 212, 379–412 consecutive angles of, 380 diagonal of, 380 general, 380 isosceles, 379 opposite angles of, 380 opposite sides of, 380 proof of as isosceles trapezoid, 403–407 as parallelogram, 385–387 as rectangle, 390 as rhombus, 394–396 as square, 399–401 Quarter turn, 240 R Radius of circle, 536 of sphere, 459 Range, 250 Ratio defined, 475–476 of similitude, 486 Rational numbers, 3 Ray, 14–15 endpoint of, 15 Real number line, distance between two points on, 8 Real numbers, 3 properties of system, 4–5 Reasoning deductive, 97, 100–103, 150–151 inductive, 94–97 Rectangle(s), 389–391 altitude of, 389 angles of, 389 base of, 389 diagonals of, 389 proof of quadrilateral as, 390 properties of, 390 Rectangular coordinates, 211 Rectangular parallelepiped, 442 Rectangular solid, 442–443 Reflection(s) glide, 243–245 line, 214–220 line of, 214 in line y x, 224–225 point, 227–231 in x-axis, 223 in y-axis, 222–223 Reflexive property, 156, 487 of equality, 110–111 Regular polygon(s), 369 center of, 450 Regular pyramid(s), 449 lateral faces of, 451 lateral sides of, 451 properties of, 450–451 Remote interior angle, 277 Replacement set, 36 Rhombus, 393–396 diagonals of, 394 proof of quadrilateral as, 394–396 properties of, 394 Riemann, Georg, 379 Right angle, 17 Right circular cone, 456 Right circular conical surface, 456 Right circular cylinder, 453 Right prism, 441 Right triangle(s), 25, 26, 515 45 |
-45 degree, 517–518 proportions in, 510–513 proving congruent by hypotenuse, leg, 362–365 Pythagorean theorem and, 515–517 30-60 degree, 518 rk, 217 RO, 230 Roots postulate, 125–126 Rotation in coordinate plane, 238–239, 240–242 defined, 238 negative, 239 positive, 239 preservation of distance under, about a fixed point, 239 Index 641 Rotational symmetry, 239–242 RP, 229 RP,d, 239 rx-axis, 223 ry-axis, 222 ryx, 224 S Saccheri, Girolamo, 379 SAS triangle congruence, 158. See also Side-angle-side triangle congruence SAS similarity theorem (SAS), 491–492. See also Side-angleside similarity theorem Scalene triangle, 24, 25, 98 Secant(s) angle formed by two intersecting, 568–571 of circle, 558 in coordinate plane, 588–591 external segment of, 576 segment formed by intersecting, 576–578 segment formed by tangent intersecting, 575–576 Segment. See Line segments Semicircle, 537 Semiperimeter, 174 Sentence closed, 37 compound, 42, 53 incomplete, 35 mathematical, 35 nonmathematical, 35–36 open, 36–37 truth value of, 35 Set, 2, 7 noncollinear, of points, 7 replacement, 36 solution, 36 truth, 36 Side(s) adjacent, 380 of angle, 15 classifying triangles according to, 24–25 consecutive, 380 corresponding, 155 inequalities involving, for triangle, 281–284 of polygon, 23 Side-angle-side similarity theorem, 491–492. See also SAS similarity theorem Side-angle-side triangle congruence, 158, See also SAS triangle congruence Side-side-side similarity theorem, 491. See also SSS similarity theorem Side-side-side triangle congruence, 165. See also SSS triangle congruence Side-side-side triangle similarity, 491. See also SSS triangle similarity Similar polygons, 486–488 14365Index.pgs 7/13/07 10:19 AM Page 642 642 Index Similarity equivalence relation of, 487–488 postulate of, 490 Similitude, ratio of, 486 Skew lines, 421–422 Sl |
ant height of cone, 456 of pyramid, 449 Slope of line, 291–294 negative, 293 of perpendicular lines, 307–310 positive, 292 undefined, 293 zero, 293 Slope-intercept form of an equation, 298 Solid geometry, 420 Solid, rectangular, 442–443 Solution set, 36 Sphere, 459–463 center of, 459 great circle of, 460 radius of, 459 surface area of, 462–463 volume of, 462–463 Square(s), 399–401 properties of, 399 proving that quadrilateral is, 399–401 Squaring of circle, 604 SSS similarity theorem (SSS), 491. See also, Side-side-side similarity theorem SSS triangle congruence, 165. See also Side-side-side triangle congruence Statement(s) compound, 42, 53 negation of, 38 postulates used in proving, 110–113 symbols and, 37 Straight angle, 16, 17 Straight line, 1, 2 Straightedge, 196, 604 Substitution postulate, 115–117, 264, 267–268 Subtraction of angles, 20–21 inequality postulates involving, 267–268 of line segments, 12–13 Subtraction postulate, 121–122 Sum of two angles, 20 Supplementary angles, 145–146, 148–149 Supplements, 146 Surface, 1 Surface area of circular cylinder, 454 of cone, 457–458 of prism, 440–443 of pyramid, 449 of sphere, 462–463 Symbols logic, 38–39 statements and, 37 Symmetric property, 156, 487 of equality, 111 Symmetry axis of, 219, 625 line, 218–220 line of, 219 point, 229 rotational, 239–242 translational, 235 Symmetry plane, 464 T Ta,b, 235 Tangent(s) angles formed by two intersecting, 567–568, 568–571 to a circle, 558 common, 559–561 common external, 560 in coordinate plane, 588 externally, 560 internally, 560 segments formed by, intersecting a secant, 575–576 Tangent segment, 561–563 Tangent to circle, 558 Terms, undefined, 1, 2 Tessellation, 259 Thales, 1 Theorem(s), 110 defined, 141 Exterior Angle, 349 Exterior Angle Inequality, 277 Hypotenuse-Leg Triangle Congru |
ence, 362–365 involving pairs of angles, 146–148 Isosceles Triangle, 181 converse of, 357–360 Midsegment, 480 Perpendicular Bisector Concurrence, 193 proving, about angles, 144–145 Pythagorean, 474, 515 converse of, 515–517 Triangle Inequality, 273 Triangle Similarity, 490 30-60-degree right triangle, 518 Total surface area of prism, 442 Transformation(s), 215 composition of, 251–252 as functions, 250–254 dilation, 247–248, 495–499 glide reflection, 243–245 line reflection, 214–220 point reflection, 227–231 rotation, 238–239, 240–242 translation, 232–235 Transformational proof, 308 Transitive property, 156, 157, 263–264, 487, 488 of equality, 111 of inequality, 264 Translation(s) of a units in the horizontal direction and b units in the vertical direction, 233 in coordinate plane, 232–235 preservation of angle measure, 232 preservation of collinearity, 232 preservation of distance, 232 preservation of midpoint, 232 defined, 232 Translational symmetry, 235 Transversal(s) defined, 330 parallel lines and, 330–332 Trapezoid, 402–407 bases of, 402 legs of, 402 median of, 405 Tree diagram, 42 Triangle(s), 23 acute, 25, 105 altitude of, 175 angle bisector of, 176 centroid of, 506 classifying according to angles, 25–26 according to sides, 24–25 concurrence of altitudes of, 317–320 concurrence of angle bisectors of, 364–365 concurrence of medians of, 506–509 congruence of, 134, 155–156, 174–203 defined, 23 equiangular, 25 equilateral, 24, 25, 181–183 exterior angles of, 277–279 included sides and included angles of, 24 inequalities involving an exterior angle of, 276–279 involving lengths of sides of, 273–274 involving sides and angles of, 281–284 isosceles, 24, 25, 181–183, 451 line segments associated with, 175–177 median of, 175–176 obtuse, 25 opposite sides and opposite angles in, 24 proportional relations among segments related to, 502–505 proving congruent by AAS, 352–355 ASA, 161–163 SAS, 158–159 SSS, 165 |
HL, 362–365 proving similar by AA, 489–494 SSS, 489–494 SAS, 489–494 right, 25, 26, 515 45-45 degree, 517–518 proportions in, 510–513 Pythagorean theorem and, 515–517 30-60 degree, 518 scalene, 24, 25, 98 sum of measures of angles of, 347–350 Triangle Inequality Theorem, 273 Triangle Similarity Theorem, 490 Triangular prism, volume of, 447 14365Index.pgs 7/13/07 10:19 AM Page 643 Trichotomy postulate, 264–265 Trisection of angle, 604 Truth set, 36 Truth table, 42–44, 51 Truth value, 35, 37, 38 for conditional p → q, 54–55 sentences and, 35 al-Tusi, Nasir al-Din, 379 Two-column proof, 101 U Undefined term, 1, 2 Upper base angles, 404 of isosceles trapezoid, 404 V Valid argument, 75, 109 Value(s) absolute, 7–8 truth, 35, 37, 38, 54–55 Vertex of angle, 15 of cone, 456 of pyramid, 449 Vertex angle of isosceles triangle, 25 Vertical angles, 149 as congruent, 150–151 Vertices adjacent, 368, 380 consecutive, 380 of polyhedron, 440 Volume of circular cylinder, 454 of cone, 457–458 of prism, 446–447 of pyramid, 449 of sphere, 462–463 of triangular prism, 447 Index 643 X x-axis, reflection in, 223 x-coordinate, 210 x-intercept, 296 (x, y), 210 Y y-axis, reflection in, 222–223 y-coordinate, 210 y-intercept, 296 Z Zero, multiplication property of, 5 Zero slope, 293 14365Index.pgs 7/13/07 10:19 AM Page 644This design is called a tessellation, or tiling, of regular hexagons. IMPROVING YOUR ALGEGRA SKILLS Algebraic Magic Squares I A magic square is an arrangement of numbers in a square grid. The numbers in every row, column, or diagonal add up to the same number. For example, in the magic square on the left, the sum of each row, column, and diagonal |
is 18. Complete the 5-by-5 magic square on the right. Use only the numbers in this list: 6, 7, 9, 13, 17, 21, 23, 24, 27, and 28. 5 4 9 10 6 2 3 8 7 20 29 22 19 12 10 25 18 11 8 14 26 30 15 16 12 CHAPTER 0 Geometric Art L E S S O N 0.4 Everything is an illusion, including this notion. STANISLAW J. LEC Op Art Op art, or optical art, is a form of abstract art that uses lines or geometric patterns to create a special visual effect. The contrasting dark and light regions sometimes appear to be in motion or to represent a change in surface, direction, and dimension. Victor Vasarely was one artist who transformed grids so that spheres seem to bulge from them. Recall the series Tsiga I, II, and III that appears in Lesson 0.1. Harlequin, shown at right, is a rare Vasarely work that includes a human form. Still, you can see Vasarely’s trademark sphere in the clown’s bulging belly In Hesitate, by contemporary op artist Bridget Riley (b 1931), what effect do the changing dots produce? In Harlequin, Victor Vasarely used curved lines and shading to create the form of a clown in motion. Op art is fun and easy to create. To create one kind of op art design, first make a design in outline. Next, draw horizontal or vertical lines, gradually varying the space between the lines to create an illusion of hills and valleys. Finally, color in alternating spaces. The Wavy Letter Step 1 Step 2 Step 3 LESSON 0.4 Op Art 13 To create the next design, first locate a point on each of the four sides of a square. Each point should be the same distance from a corner, as shown. Your compass is a good tool for measuring equal lengths. Connect these four points to create another square within the first. Repeat the process until the squares appear to converge on the center. Be careful that you don’t fall in! The Square Spiral Step 1 Step 2 Step 3 Step 4 Here are some other examples of op art. Square tunnel or top of pyramid? Amish quilt, tumbling block design Japanese Op Art, Hajime Juchi, Dover Publications Op art by Carmen Apodaca, geometry student You can create any of the designs on this page using just |
a compass and straightedge (and doing some careful coloring). Can you figure out how each of these op art designs was created? 14 CHAPTER 0 Geometric Art EXERCISES 1. What is the optical effect in each piece of art in this lesson? 2. Nature creates its own optical art. At first the black and white stripes of a zebra appear to work against it, standing out against the golden brown grasses of the African plain. However, the stripes do provide the zebras with very effective protection from predators. When and how? 3. Select one type of op art design from this lesson and create your own version of it. 4. Create an op art design that has reflectional symmetry, but not rotational symmetry. 5. Antoni Gaudí (1852–1926) designed the Bishop’s Palace in Astorga, Spain. List as many geometric shapes as you can recognize on the palace (flat, two-dimensional shapes such as rectangles as well as solid, threedimensional shapes such as cylinders). What type of symmetry do you see on the palace? Bishop’s Palace, Astorga, Spain IMPROVING YOUR REASONING SKILLS Bagels In the original computer game of bagels, a player determines a three-digit number (no digit repeated) by making educated guesses. After each guess, the computer gives a clue about the guess. Here are the clues. bagels: no digit is correct pico: one digit is correct but in the wrong position fermi: one digit is correct and in the correct position In each of the games below, a number of guesses have been made, with the clue for each guess shown to its right. From the given set of guesses and clues, determine the three-digit number. If there is more than one solution, find them all. Game 1??? bagels pico pico pico fermi pico Game 2??? bagels pico pico fermi fermi pico pico LESSON 0.4 Op Art 15 Knot Designs Knot designs are geometric designs that appear to weave or to interlace like a knot. Some of the earliest known designs are found in Celtic art from the northern regions of England and Scotland. In their carved stone designs, the artists imitated the rich geometric patterns of three-dimensional crafts such as weaving and basketry. The Book of Kells (8th and 9th centuries) is the most famous |
collection of Celtic knot designs. L E S S O N 0.5 In the old days, a love-sick sailor might send his sweetheart a length of fishline loosely tied in a love knot. If the knot was returned pulled tight it meant the passion was strong. But if the knot was returned untied— ah, matey, time to ship out. OLD SAILOR’S TALE Carved knot pattern from Nigeria Celtic knot design Today a very familiar knot design is the set of interconnected rings (shown at right) used as the logo for the Olympic Games. Here are the steps for creating two examples of knot designs. Look them over before you begin the exercises. Step 1 Step 2 Step 3 Step 4 You can use a similar approach to create a knot design with rings. Step 1 Step 2 Step 3 Step 4 16 CHAPTER 0 Geometric Art Here are some more examples of knot designs. Knot design by Scott Shanks, geometry student Tiger Tail, Diane Cassell, parent of geometry student Medieval Russian knot design Japanese knot design The last woodcut made by M. C. Escher is a knot design called Snakes. The rings and the snakes interlace, and the design has 3-fold rotational symmetry. Snakes, M. C. Escher, 1969/ ©2002 Cordon Art B. V.–Baarn–Holland. All rights reserved. LESSON 0.5 Knot Designs 17 EXERCISES 1. Name a culture or country whose art uses knot designs. You will need Construction tools for Exercise 3 2. Create a knot design of your own, using only straight lines on graph paper. 3. Create a knot design of your own with rotational symmetry, using a compass or a circle template. 4. Sketch five rings linked together so that you could separate all five by cutting open one ring. 5. The coat of arms of the Borromeo family, who lived during the Italian Renaissance (ca. 15th century), showed a very interesting knot design known as the Borromean Rings. In it, three rings are linked together so that if any one ring is removed the remaining two rings are no longer connected. Got that? Good. Sketch the Borromean Rings. 6. The Chokwe storytellers of northeastern Angola are called Akwa kuta sona (“those who know how to draw”). When they sit down to draw and to tell their stories, they clear the ground and set up a grid of |
points in the sand with their fingertips, as shown below left. Then they begin to tell a story and, at the same time, trace a finger through the sand to create a lusona design with one smooth, continuous motion. Try your hand at creating sona (plural of lusona). Begin with the correct number of dots. Then, in one motion, re-create one of the sona below. The initial dot grid is shown for the rat. Initial dot grid Rat Mbemba bird Scorpion 7. In Greek mythology, the Gordian knot was such a complicated knot that no one could undo it. Oracles claimed that whoever could undo the knot would become the ruler of Gordium. When Alexander the Great (356–323 B.C.E.) came upon the knot, he simply cut it with his sword and claimed he had fulfilled the prophecy, so the throne was his. The expression “cutting the Gordian knot” is still used today. What do you think it means? Science Mathematician DeWitt Sumners at Florida State University and biophysicist Sylvia Spenger at the University of California, Berkeley, have discovered that when a virus attacks DNA, it creates a knot on the DNA. 18 CHAPTER 0 Geometric Art 8. The square knot and granny knot are very similar but do very different things. Compare their symmetries. Use string to re-create the two knots and explain their differences. Square knot Granny knot 9. Cut a long strip of paper from a sheet of lined paper or graph paper. Tie the strip of paper snugly, but without wrinkles, into a simple knot. What shape does the knot create? Sketch your knot. SYMBOLIC ART Japanese artist Kunito Nagaoka (b 1940) uses geometry in his work. Nagaoka was born in Nagano, Japan, and was raised near the active volcano Asama. In Japan, he experienced earthquakes and typhoons as well as the human tragedies of Hiroshima and Nagasaki. In 1966, he moved to Berlin, Germany, a city rebuilt in concrete from the ruins of World War II. These experiences clearly influenced his work. You can find other examples of symbolic art at www.keymath.com/DG. Look at the etching shown here, or another piece of symbolic art. Write a paragraph describing what you think might have happened in the scene or what you think it might represent. What types of geometric figures do you find? Use geometric shapes in your |
own sketch or painting to evoke a feeling or to tell a story. Write a one- or two-page story related to your art. ISEKI/PY XVIII (1978), Kunito Nagaoka LESSON 0.5 Knot Designs 19 Islamic Tile Designs Islamic art is rich in geometric forms. Early Islamic, or Muslim, artists became familiar with geometry through the works of Euclid, Pythagoras, and other mathematicians of antiquity, and they used geometric patterns extensively in their art and architecture. L E S S O N 0.6 Patience with small details makes perfect a large work, like the universe. JALALUDDIN RUMI An exterior wall of the Dome of the Rock (660–750 C.E.) mosque in Jerusalem Islam forbids the representation of humans or animals in religious art. So, instead, the artists use intricate geometric patterns. One of the most striking examples of Islamic architecture is the Alhambra Palace, in Granada, Spain. Built over 600 years ago by Moors and Spaniards, the Alhambra is filled from floor to ceiling with marvelous geometric patterns. The designs you see on this page are but a few of the hundreds of intricate geometric patterns found in the tile work and the inlaid wood ceilings of buildings like the Alhambra and the Dome of the Rock. Carpets and hand-tooled bronze plates from the Islamic world also show geometric designs. The patterns often elaborate on basic grids of regular hexagons, equilateral triangles, or squares. These complex Islamic patterns were constructed with no more than a compass and a straightedge. Repeating patterns like these are called tessellations. You’ll learn more about tessellations in Chapter 7. Alcove in the Hall of Ambassadors, the Alhambra, a Moorish palace in Granada, Spain 20 CHAPTER 0 Geometric Art The two examples below show how to create one tile in a square-based and a hexagon-based design. The hexagon-based pattern is also a knot design. 8-pointed Star Step 1 Step 2 Step 3 Step 4 Step 5 Hexagon Tile Design Step 1 Step 2 Step 3 Step 4 Step 5 LESSON 0.6 Islamic Tile Designs 21 In Morocco, zillij, the art of using glazed tiles to form geometric patterns, is the most common practice for making mosaics. Zillij artists cut stars, octagons, and other shapes from clay tiles and place them upside |
down into the lines of their design. When the tiling is complete, artists pour concrete over the tiles to form a slab. When the concrete dries, they lift the whole mosaic, displaying the colors and connected shapes, and mount it against a fountain, palace, or other building. EXERCISES 1. Name two countries where you can find Islamic architecture. 2. What is the name of the famous palace in Granada, Spain, where you can find beautiful examples of tile patterns? You will need Construction tools for Exercises 5–7 3. Using tracing paper or transparency film, trace a few tiles from the 8-pointed star design. Notice that you can slide, or translate, the tracing in a straight line horizontally, vertically, and even diagonally to other positions so that the tracing will fit exactly onto the tiles again. What is the shortest translation distance you can find, in centimeters? 4. Notice that when you rotate your tracing from Exercise 3 about certain points in the tessellation, the tracing fits exactly onto the tiles again. Find two different points of rotation. (Put your pencil on the point and try rotating the tracing paper or transparency.) How many times in one rotation can you make the tiles match up again? 22 CHAPTER 0 Geometric Art Architecture After studying buildings in other Muslim countries, the architect of the Petronas Twin Towers, Cesar Pelli (b 1926), decided that geometric tiling patterns would be key to the design. For the floor plan, his team used a very traditional tile design, the 8-pointed star—two intersecting squares. To add space and connect the design to the traditional “arabesques,” the design team added arcs of circles between the eight points. 5. Currently the tallest buildings in the world are the Petronas Twin Towers in Kuala Lumpur, Malaysia. Notice that the floor plans of the towers have the shape of Islamic designs. Use your compass and straightedge to re-create the design of the base of the Petronas Twin Towers, shown at right. 6. Use your protractor and ruler to draw a square tile. Use your compass, straightedge, and eraser to modify and decorate it. See the example in this lesson for ideas, but yours can be different. Be creative! 7. Construct a regular hexagon tile and modify and decorate it. See the example in this lesson for ideas, but yours can be different. 8. Create a tessellation with |
one of the designs you made in Exercises 6 and 7. Trace or photocopy several copies and paste them together in a tile pattern. (You can also create your tessellation using geometry software and print out a copy.) Add finishing touches to your tessellation by adding, erasing, or whiting out lines as desired. If you want, see if you can interweave a knot design within your tessellation. Color your tessellation. PHOTO OR VIDEO SAFARI In Lesson 0.1, you saw a few examples of geometry and symmetry in nature and art. Now go out with your group and document examples of geometry in nature and art. Use a camera or video camera to take pictures of as many examples of geometry in nature and art as you can. Look for many different types of symmetry, and try to photograph art and crafts from many different cultures. Consider visiting museums and art galleries, but make sure it’s okay to take pictures when you visit. You might find examples in your home or in the homes of friends and neighbors. If you take photographs, write captions for them that describe the geometry and the types of symmetries you find. If you record video, record your commentary on the soundtrack. LESSON 0.6 Islamic Tile Designs 23 ● CHAPTER 11 REVIEW ● CHAPTER 0 REVIEW ● CHAPTER 0 REVIEW ● CHAPTER 0 REVIEW ● CHAPTER CHAPTER 0 R E V I E W In this chapter, you described the geometric shapes and symmetries you see in nature, in everyday objects, in art, and in architecture. You learned that geometry appears in many types of art—ancient and modern, from every culture—and you learned specific ways in which some cultures use geometry in their art. You also used a compass and straightedge to create your own works of geometric art. The end of a chapter is a good time to review and organize your work. Each chapter in this book will end with a review lesson. EXERCISES 1. List three cultures that use geometry in their art. You will need Construction tools for Exercises 4, 5, and 10 2. What is the optical effect of the op art design at right? 3. Name the basic tools of geometry you used in this chapter and describe their uses. 4. With a compass, draw a 12-petal daisy. 5. Construction With a compass and straightedge, construct a regular hexagon. 6. List |
three things in nature that have geometric shapes. Name their shapes. 7. Draw an original knot design. 8. Which of the wheels below have reflectional symmetry? Hot Blocks (1966–67), Edna Andrade How many lines of symmetry does each have? Wheel A Wheel B Wheel C Wheel D 9. Which of the wheels in Exercise 8 have only rotational symmetry? What kind of rotational symmetry does each have? 24 CHAPTER 0 Geometric Art ● CHAPTER 0 REVIEW ● CHAPTER 0 REVIEW ● CHAPTER 0 REVIEW ● CHAPTER 0 REVIEW ● CHAPTER 0 10. A mandala is a circular design arranged in rings that radiate from the center. (See the Cultural Connection below.) Use your compass and straightedge to create a mandala. Draw several circles using the same point as the center. Create a geometric design in the center circle, and decorate each ring with a symmetric geometric design. Color or decorate your mandala. Two examples are shown below. The first mandala uses daisy designs. The second mandala is a combination knot and Islamic design by Scott Shanks, geometry student. 11. Create your own personal mandala. You might include your name, cultural symbols, photos of friends and relatives, and symbols that have personal meaning for you. Color it. 12. Create one mandala that uses techniques from Islamic art, is a knot design, and also has optical effects. Cultural The word mandala comes from Sanskrit, the classical language of India, and means “circle” or “center.” Hindus use mandala designs for meditation. The Aztec calendar stone below left is an example of a mandala. In the center is the mask of the sun god. Notice the symbols are arranged symmetrically within each circle. The rose windows in many gothic cathedrals, like the one below right from the Chartres Cathedral in France, are also mandalas. Notice all the circles within circles, each one filled with a design or picture. CHAPTER 0 REVIEW 25 EW ● CHAPTER 0 REVIEW ● CHAPTER 0 REVIEW ● CHAPTER 0 REVIEW ● CHAPTER 0 REVIEW ● CHAPTE 13. Did you know that “flags” is the most widely read topic of the World Book Encyclopedia? Research answers to these questions. More information about flags is available at www.keymath.com/DG. a. Is the flag of Puerto Rico symmetric? Explain. b. |
Does the flag of Kenya have rotational symmetry? Explain. c. Name a country whose flag has both rotational and reflectional symmetry. Sketch the flag. Assessing What You’ve Learned KEEPING A PORTFOLIO This section suggests how you might review, organize, and communicate to others what you’ve learned. Whether you follow these suggestions or directions from your teacher, or use study strategies of your own, be sure to reflect on all you’ve learned. An essential part of learning is being able to show yourself and others how much you know and what you can do. Assessment isn’t limited to tests and quizzes. Assessment isn’t even limited to what your teacher sees or what makes up your grade. Every piece of art you make, and every project or exercise you complete, gives you a chance to demonstrate to somebody—yourself, at least—what you’re capable of. BEGIN A PORTFOLIO This chapter is primarily about art, so you might organize your work the way a professional artist does—in a portfolio. A portfolio is different from a notebook, both for an artist and for a geometry student. An artist’s notebook might contain everything from scratch work to practice sketches to random ideas jotted down. A portfolio is reserved for an artist’s most significant or best work. It’s his or her portfolio that an artist presents to the world to demonstrate what he or she is capable of doing. The portfolio can also show how an artist’s work has changed over time. Review all the work you’ve done so far and choose one or more examples of your best art projects to include in your portfolio. Write a paragraph or two about each piece, addressing these questions: What is the piece an example of? Does this piece represent your best work? Why else did you choose it? What mathematics did you learn or apply in this piece? How would you improve the piece if you redid or revised it? Portfolios are an ongoing and ever-changing display of your work and growth. As you finish each chapter, update your portfolio by adding new work. 26 CHAPTER 0 Geometric Art CHAPTER 1 Introducing Geometry Although I am absolutely without training or knowledge in the exact sciences, I often seem to have more in common with mathematicians than with my fellow artists. M. C. ESCHER Three Worlds, M. C. Escher, 1955 ©2002 Cordon Art |
B. V.–Baarn–Holland. All rights reserved In this chapter you will ● write your own definitions of many geometry terms and geometric figures ● start a notebook with a list of all the terms and their definitions ● develop very useful visual thinking skills L E S S O N 1.1 Nature’s Great Book is written in mathematical symbols. GALILEO GALILEI Building Blocks of Geometry Three building blocks of geometry are points, lines, and planes. A point is the most basic building block of geometry. It has no size. It has only location. You represent a point with a dot, and you name it with a capital letter. The point shown below is called P. A tiny seed is a physical model of a point. P Mathematical model of a point A line is a straight, continuous arrangement of infinitely many points. It has infinite length but no thickness. It extends forever in two directions. You name a line by giving the letter names of any two points on the line and by placing the line symbol above the letters, for example, AB or BA. B A A piece of spaghetti is a physical model of a line. A line, however, is longer, straighter, and thinner than any piece of spaghetti ever made. Mathematical model of a line A plane has length and width but no thickness. It is like a flat surface that extends infinitely along its length and width. You represent a plane with a four-sided figure, like a tilted piece of paper, drawn in perspective. Of course, this actually illustrates only part of a plane. You name a plane with a script capital letter, such as. A flat piece of rolledout dough is a model of a plane, but a plane is broader, wider, and thinner than any piece of dough you could roll. Mathematical model of a plane 28 CHAPTER 1 Introducing Geometry Investigation Mathematical Models Step 1 Step 2 Identify examples of points, lines, and planes in these pictures. Explain in your own words what point, line, and plane mean. It can be difficult to explain what points, lines, and planes are. Yet, you probably recognized several models of each in the investigation. Early mathematicians tried to define these terms. By permission of Johnny Hart and Creators Syndicate, Inc. LESSON 1.1 Building Blocks of Geometry 29 The ancient Greeks said, “A point is that which has no part. A line is breadthless length.” The Mohist philosophers of |
ancient China said, “The line is divided into parts, and that part which has no remaining part is a point.” Those definitions don’t help much, do they? A definition is a statement that clarifies or explains the meaning of a word or a phrase. However, it is impossible to define point, line, and plane without using words or phrases that themselves need definition. So these terms remain undefined. Yet, they are the basis for all of geometry. Using the undefined terms point, line, and plane, you can define all other geometry terms and geometric figures. Many are defined in this book, and others will be defined by you and your classmates. Keep a definition list in your notebook, and each time you encounter new geometry vocabulary, add the term to your list. Illustrate each definition with a simple sketch. Here are your first definitions. Begin your list and draw sketches for all definitions. Collinear means on the same line. A B C Points A, B, and C are collinear. Coplanar means on the same plane. D E F Points D, E, and F are coplanar. Name three balls that are collinear. Name three balls that are coplanar but not collinear. Name four balls that are not coplanar. B A C D F E G 30 CHAPTER 1 Introducing Geometry A line segment consists of two points called the endpoints of the segment and all the points between them that are collinear with the two points. Line segment B A Endpoints You can write line segment AB, using a segment symbol, as AB or BA. There are two ways to write the length of a segment. You can write AB 2 in., meaning the distance from A to B is 2 inches. You can also use an m for “measure” in front of the segment name, and write the distance as mAB 2 in. If no measurement units are used for the length of a segment, it is understood that the choice of units is not important, or is based on the length of the smallest square in the grid. Figure A A 2 B Figure B y M 4 5 4 N x AB 2 in., or mAB 2 in. mMN 5 units Two segments are congruent segments if and only if they have the same measure or length. The symbol for congruence is, and you say it as “is congruent to.” You use the equals symbol, |
, between equal numbers and the congruence symbol,, between congruent figures. C 3.2 cm 3.2 cm A D AC DC AC DC When drawing figures, you show congruent segments by making identical markings. These single marks mean these two segments are congruent to each other. B A C S P These double marks SP RQ, mean that and these triple marks mean that PQ SR. R Q The midpoint of a segment is the point on the segment that is the same distance from both endpoints. The midpoint bisects the segment, or divides the segment into two congruent segments. LESSON 1.1 Building Blocks of Geometry 31 EXAMPLE Study the diagrams below. a. Name each midpoint and the segment it bisects. b. Name all the congruent segments. Use the congruence symbol to write your answers. C 2 cm G E F 2 cm H D J K L M N P Solution Look carefully at the markings and apply the midpoint definition. a. CF FD, so F is the midpoint of CD; JK KL, so K is the midpoint of JL. b. CF FD, HJ HL, and JK KL. Even though EF and FG appear to have the same length, you cannot assume they are congruent without the markings. The same is true for MN and NP. Ray AB is the part of AB that contains point A and all the points on AB that are on the same side of point A as point B. Imagine cutting off all the points to the left of point A. Endpoint Ray AB A Y B A Y B In the figure above, AY and AB are two ways to name the same ray. Note that AB is not the same as BA! A ray begins at a point and extends infinitely in one direction. You need two letters to name a ray. The first letter is the endpoint of the ray, and the second letter is any other point that the ray passes through. AB BA B B A A Physical model of a ray: beams of light 32 CHAPTER 1 Introducing Geometry EXERCISES 1. Identify the models in the photos below for point, segment, plane, collinear points, and coplanar points. For Exercises 2–4, name each line in two different ways. 2. P T 3. A R T 4. y 5 M SA x 8 For Exercises 5– |
7, draw two points and label them. Then use a ruler to draw each line. Don’t forget to use arrowheads to show that it extends indefinitely. 5. AB 6. KL 7. DE with D(3, 0) and E(0, 3) For Exercises 8–10, name each line segment. A 8. C 9. y 5 P 10. R I Q x 10 T For Exercises 11 and 12, draw and label each line segment. 11. AB 12. RS with R(0, 3) and S(2, 11) For Exercises 13 and 14, use your ruler to find the length of each line segment to the nearest tenth of a centimeter. Write your answer in the form mAB?. A C 13. 14. D B For Exercises 15–17, use your ruler to draw each segment as accurately as you can. Label each segment. 15. AB 4.5 cm 16. CD 3 in. 17. EF 24.8 cm LESSON 1.1 Building Blocks of Geometry 33 18. Name each midpoint and the segment it bisects 19. Draw two segments that have the same midpoint. Mark your drawing to show congruent segments. 20. Draw and mark a figure in which M is the midpoint of ST, SP PT, and T is the midpoint of PQ. For Exercises 21–23, name the ray in two different ways. 21. A B C 22. M N P 23. Z Y X For Exercises 24–26, draw and label each ray. 25. YX 24. AB 26. MN 27. Draw a plane containing four coplanar points A, B, C, and D, with exactly three collinear points A, B, and D. 28. Given two points A and B, there is only one segment that you can name: AB. With three collinear points A, B, and C, there are three different segments that you can name: AB, AC, and BC. With five collinear points A, B, C, D, and E, how many different segments can you name? For Exercises 29–31, draw axes onto graph paper and locate point A(4, 0) as shown. 29. Draw AB, where point B has coordinates (2, 6). 30. Draw OM with endpoint (0, 0) that goes through point M |
(2, 2). 31. Draw CD through points C(2, 1) and D(2, 3). y 5 –5 x A (4, 0) –5 Career Woodworkers use a tool called a plane to shave a rough wooden surface to create a perfectly smooth planar surface. The smooth board can then be made into a tabletop, a door, or a cabinet. Woodworking is a very precise process. Producing high-quality pieces requires an understanding of lines, planes, and angles as well as careful measurements. 34 CHAPTER 1 Introducing Geometry 32. If the signs of the coordinates of collinear points P(6, 2), Q(5, 2), and R(4, 6) are reversed, are the three new points still collinear? Draw a picture and explain why. 33. Draw a segment with midpoint N(3, 2). Label it PQ. 34. Copy triangle TRY shown at right. Use your ruler to find the midpoint A of side TR and the midpoint G of side TY. Draw AG. T R Y SPIRAL DESIGNS The circle design shown below is used in a variety of cultures to create mosaic decorations. The spiral design may have been inspired by patterns in nature. Notice that the seeds on the sunflower also spiral out from the center. Here are the steps to make the spirals. Step 1 Step 2 Step 3 Step 4 The more circles and radii you draw, the more detailed your design will be. Create and decorate your own spiral design. LESSON 1.1 Building Blocks of Geometry 35 ALGEBRA SKILLS 1 ● USING YOUR ALGEBRA SKILLS 1 ● USING YOUR ALGEBRA SKILLS 1 ● USING YO USING YOUR ALGEBRA SKILLS 1 Midpoint A midpoint is the point on a line segment that is the same distance from both endpoints. You can think of a midpoint as being halfway between two locations. You know how to mark a midpoint. But when the position and location matter, such as in navigation and geography, you can use a coordinate grid and some algebra to find the exact location of the midpoint. You can calculate the coordinates of the midpoint of a segment on a coordinate grid using a formula. Coordinate Midpoint Property If x1, y1 then the coordinates of the midpoint are and x2, y2 are the coordinates of the end |
points of a segment, x1 x2, y1 2 2 y2 History Surveyors and mapmakers of ancient Egypt, China, Greece, and Rome used various coordinate systems to locate points. Egyptians made extensive use of square grids and used the first known rectangular coordinates at Saqqara around 2650 B.C.E. By the seventeenth century, the age of European exploration, the need for accurate maps and the development of easy-to-use algebraic symbols gave rise to modern coordinate geometry. Notice the lines of latitude and longitude in this seventeenth-century map. 36 CHAPTER 1 Introducing Geometry ALGEBRA SKILLS 1 ● USING YOUR ALGEBRA SKILLS 1 ● USING YOUR ALGEBRA SKILLS 1 ● USING YO EXAMPLE Solution Segment AB has endpoints (8, 5) and (3, 6). Find the coordinates of the midpoint of AB. y The midpoint is not on a grid intersection point, so we can use the coordinate midpoint property. 3) 2.5 x2 (8 x x1 2 2 14) 0.5 y2 (13 y y1 2 2 The midpoint of AB is (2.5, 0.5). A (–8, 5) (–2.5, –0.5) x B (3, –6) EXERCISES For Exercises 1–3, find the coordinates of the midpoint of the segment with each pair of endpoints. 1. (12, 7) and (6, 15) 2. (17, 8) and (1, 11) 3. (14, 7) and (3, 18) 4. One endpoint of a segment is (12, 8). The midpoint is (3, 18). Find the coordinates of the other endpoint. 5. A classmate tells you, “Finding the coordinates of a midpoint is easy. You just find the averages.” Is there any truth to it? Explain what you think your classmate means. 6. Find the two points on AB that divide the segment into three congruent parts. Point A has coordinates (0, 0) and point B has coordinates (9, 6). Explain your method. 7. Describe a way to find points that divide a segment into fourths. 8. In each figure below, imagine drawing the diagonals AC and BD. a. Find the midpoint of AC and |
the midpoint of BD in each figure. b. What do you notice about the midpoints? y A B B 10 5 B C (35.5, 10) C (22, 8.5) Figure 1 Figure 2 Figure 3 A D 5 C 10 D (16, 0.5) A 15 20 25 D (29.5, 1) 30 35 x USING YOUR ALGEBRA SKILLS 1 Midpoint 37 L E S S O N 1.2 Inspiration is needed in geometry, just as much as in poetry. ALEKSANDR PUSHKIN Poolroom Math People use angles every day. Plumbers measure the angle between connecting pipes to make a good fitting. Woodworkers adjust their saw blades to cut wood at just the correct angle. Air traffic controllers use angles to direct planes. And good pool players must know their angles to plan their shots. Is the angle between the two hands of the wristwatch smaller than the angle between the hands of the large clock? “Little Benji,” the wristwatch Big Ben at the Houses of Parliament in London, England 38 CHAPTER 1 Introducing Geometry You can use the terms that you defined in Lesson 1.1 to write a precise definition of angle. An angle is formed by two rays that share a common endpoint, provided that the two rays are noncollinear. In other words, the rays cannot lie on the same line. The common endpoint of the two rays is the vertex of the angle. The two rays are the sides of the angle. You can name the angle in the figure below angle TAP or angle PAT, or use the angle symbol and write TAP or PAT. Notice that the vertex must be the middle letter, and the first and last letters each name a point on a different ray. Since there are no other angles with vertex A, you can also simply call this A. T Vertex A Sides P Career In sports medicine, specialists may examine the healing rate of an injured joint by its angle of recovery. For example, a physician may assess how much physical therapy a patient needs by measuring the degree to which a patient can bend his or her ankle from the floor. EXAMPLE A Name all the angles in these drawings. Solution The angles are T, V, TUV, 1, TUR, XAY, YAZ, and XAZ. (Did you get them all?) Notice that 1 is a shorter way to name RUV Which angles in Example A seem |
big to you? Which seem small? The measure of an angle is the smallest amount of rotation about the vertex from one ray to the other, measured in degrees. According to this definition, the measure of an angle can be any value between 0° and 180°. The smallest amount of rotation PG to PS from is 136°. 136° P 224° G S The geometry tool you use to measure an angle is a protractor. Here’s how you use it. Step 1: Place the center mark of the protractor on the vertex. Step 2: Line up the 0-mark with one side of the angle. Step 3: Read the measure on the protractor scale. 0 30 40 5 0 6 150 140 television antenna is a physical model of an angle. Note that changing the length of the antenna doesn’t change the angle 90 100 110 12 80 70 1 0 0 90 0 1 3 0 60 Step 4: Be sure you read the scale that has the 0-mark you are using! The angle in the diagram measures 34° and not 146°. LESSON 1.2 Poolroom Math 39 To show the measure of an angle, use an m before the angle symbol. For example, mZAP 34° means the measure of ZAP is 34 degrees. EXAMPLE B Use your protractor to measure these angles as accurately as you can. Which ones measure more than 90°? Z 34° A P mZAP 34° 3 1 2 Solution Measuring to the nearest degree, you should get these approximate answers. (The symbol means “is approximately equal to.”) m3 92° 2 and 3 measure more than 90°. m1 16° m2 164° Two angles are congruent angles if and only if they have the same measure. You use identical markings to show that two angles in a figure are congruent. D G O These markings mean that DOG CAT and mDOG mCAT. A C T A ray is the angle bisector if it contains the vertex and divides the angle into two congruent angles. In the figure at right, CD bisects ACB so that ACD BCD. C B D A Science Earth takes 365.25 days to travel a full 360° around the Sun. That means that each day, Earth travels a little less than 1° in its orbit around the Sun. Meanwhile, Earth also completes one full rotation each day, making the Sun appear to rise and set |
. By how many degrees does the Sun’s position in the sky change every hour? 40 CHAPTER 1 Introducing Geometry EXAMPLE C Look for angle bisectors and congruent angles in the figures below. a. Name each angle bisector and the angle it bisects. b. Name all the congruent angles in the figure. Use the congruence symbol and name the angles so there is no confusion about which angle you mean. E 44° 43° F G H P 52° 52° Q S R M Y E N O Solution a. Use the angle bisector definition. SRP PRQ, so RP bisects SRQ. b. SRP PRQ and YMN OME. Investigation Virtual Pool Pocket billiards, or pool, is a game of angles. When a ball bounces off the pool table’s cushion, its path forms two angles with the edge of the cushion. The incoming angle is formed by the cushion and the path of the ball approaching the cushion. Outgoing angle Cushion Incoming angle The outgoing angle is formed by the cushion and the path of the ball leaving the cushion. As it turns out, the measure of the outgoing angle equals the measure of the incoming angle. Computer scientist Nesli O’Hare is also a professional pool player. LESSON 1.2 Poolroom Math 41 Use your protractor to study these shots Step 1 Step 2 Step 3 Step 4 Step 5 Use your protractor to find the measure of 1. Which is the correct outgoing angle? Which point—A or B—will the ball hit? Which point on the cushion—W, X, or Y—should the white ball hit so that the ray of the outgoing angle passes through the center of the 8-ball? Compare your results with your group members’ results. Does everyone agree? How would you hit the white ball against the cushion so that the ball passes over the same spot on the way back? How would you hit the ball so that it bounces off three different points on the cushions, without ever touching cushion CP? EXERCISES 1. Name each angle in three E T different ways. N F U 1 O 2 R For Exercises 2–4, draw and label each angle. 2. TAN 3. BIG 4. SML 5. For each figure at right, list the angles that you can name using only the vertex letter. S R T P Q 42 CHAPTER 1 |
Introducing Geometry B C D E F 6. Draw a figure that contains at least three angles and requires three letters to name each angle. For Exercises 7–14, find the measure of each angle. B C Y 0 30 40 5 0 6 150 140 90 100 110 12 80 70 1 0 0 90 0 1 3 0 60. mAQB? 8. mAQC? 9. mXQA? 10. mAQY? 11. mZQY? 12. mZQX? 13. mCQB? 14. mXQY? For Exercises 15–19, use your protractor to find the measure of the angle to the nearest degree. 15. mMAC? M C A 17. mS? 18. mSON? 19. mNOR? 16. mIBM? I N B M S O Q R 20. Which angle below has the greater measure, SML or BIG? Why? S M L I B G LESSON 1.2 Poolroom Math 43 For Exercises 21–23, use your protractor to draw angles with these measures. Label them. 21. mA 44° 22. mB 90° 23. mCDE 135° 24. Use your protractor to draw the angle bisector of A in Exercise 21 and the angle bisector of D in Exercise 23. Use markings to show that the two halves are congruent. 25. Copy triangle CAN shown at right. Use your protractor to find the angle bisector of A. Label the point where it crosses CN point Y. Use your ruler to find the midpoint of CN and label it D. Are D and Y the same point? C A For Exercises 26–28, draw a clock face with hands to show these times. 26. 3:30 27. 3:40 28. 3:15 29. Give an example of a time when the angle made by the hands of the clock will be greater than 90°. For Exercises 30–33, copy each figure and mark it with all the given information. 30. TH 6 mTHO 90° OH 8 H 31. RA SA mT mH RT SH T H N T O R A S 32. AT AG AI AN AGT ATG GI TN 33. BW TI WO IO WBT ITB BWO TIO For Exercises 34 and 35, write down what you know from the markings. Do not |
use your protractor or your ruler. 34. MI? IC? mM? Y E K M C I 35. MEO? SUE? OU? S U E M O 44 CHAPTER 1 Introducing Geometry For Exercises 36–38, do not use a protractor. Recall from Chapter 0 that a complete rotation around a point is 360°. Find the angle measures represented by each letter. 36. 37. 38. 15° 21° x 1 4 rotation y 41° 37° 1 2 rotation z 74° 135° 87° 39. If the 4-ball is hit as shown, will it go into the corner pocket? Find the path of the ball using only your protractor and straightedge. 40. What is the measure of the incoming angle? Which point will the ball pass through? Use your protractor to find out. D C B A 41. The principle you just learned for billiard balls is also true for a ray of light reflecting from a mirror. What you “see” in the mirror is actually light from an object bouncing off the mirror and traveling to your eye. Will you be able to see your shoes in this mirror? Copy the illustration and draw rays to show the light traveling from your shoes to the mirror and back to your eye. LESSON 1.2 Poolroom Math 45 Review 42. Use your ruler to draw a segment with length 12 cm. Then use your ruler to locate the midpoint. Label and mark the figure. 43. The balancing point of an object is called its center of gravity. Where is the center of gravity of a thin, rodlike piece of wire or tubing? Copy the thin wire shown below onto your paper. Mark the balance point or center of gravity. 44. Explain the difference between MS DG and MS DG. IMPROVING YOUR VISUAL THINKING SKILLS Coin Swap I and II 1. Arrange two dimes and two pennies on a grid of five squares, as shown. Your task is to switch the position of the two dimes and two pennies in exactly eight moves. A coin can slide into an empty square next to it, or it can jump over one coin into an empty space. Record your solution by drawing eight diagrams that show the moves. 2. Arrange three dimes and three pennies on a grid of seven squares, as shown. Follow the same rules as above to switch the position of the three dimes and three pennies in exactly 15 moves. Record your |
solution by listing in order which coin is moved. For example, your list might begin PDP.... 46 CHAPTER 1 Introducing Geometry L E S S O N 1.3 “When I use a word,” Humpty replied in a scornful tone,“it means just what I choose it to mean—neither more nor less.” “The question is,” said Alice,“whether you can make a word mean so many different things.” LEWIS CARROLL What’s a Widget? Good definitions are very important in geometry. In this lesson you will write your own geometry definitions. Which creatures in the last group are Widgets? A C B D Widgets Not Widgets Who are Widgets? You might have asked yourself, “What things do all the Widgets have in common, and what things do Widgets have that others do not have?” In other words, what characteristics make a Widget a Widget? They all have colorful bodies with nothing else inside; two tails—one like a crescent moon, the other like an eyeball. By observing what a Widget is and what a Widget isn’t, you identified the characteristics that distinguish a Widget from a non-Widget. Based on these characteristics, you should have selected A as the only Widget in the last group. This same process can help you write good definitions of geometric figures. This statement defines a protractor: “A protractor is a geometry tool used to measure angles.” First, you classify what it is (a geometry tool), then you say how it differs from other geometry tools (it is the one you use to measure angles). What should go in the blanks to define a square? A square is a that. Classify it. What is it? How does it differ from others? Once you’ve written a definition, you should test it. To do this, you look for a counterexample. That is, try to create a figure that fits your definition but isn’t what you’re trying to define. If you can come up with a counterexample for your definition, you don’t have a good definition. EXAMPLE A Everyone knows, “A square is a figure with four equal sides.” What’s wrong with this definition? a. Sketch a counterexample. (You |
can probably find more than one!) b. Write a better definition for a square. LESSON 1.3 What’s a Widget? 47 Solution You probably noticed that “figure” is not specific enough to classify a square, and that “four equal sides” does not specify how it differs from the first counterexample shown below. a. Three counterexamples are shown here, and you may have found others, too. b. One better definition is “A square is a 4-sided figure that has all sides congruent and all angles measuring 90 degrees.” Beginning Steps to Creating a Good Definition 1. Classify your term. What is it? (“A square is a 4-sided figure...”) 2. Differentiate your term. How does it differ from others in that class? (“... that has four congruent sides and four right angles.”) 3. Test your definition by looking for a counterexample. Ready to write a couple of definitions? First, here are two more types of markings that are very important in geometry. A restaurant counter example The same number of arrow marks indicates that lines are parallel. The symbol means “is parallel to.” A small square in the corner of an angle indicates that it measures 90°. The symbol means “is perpendicular to.” EXAMPLE B Define these terms: a. Parallel lines b. Perpendicular lines k i j i j k Solution Following these steps, classify and differentiate each term. Classify. Differentiate. a. Parallel lines are lines in the same plane that never meet. b. Perpendicular lines are lines that meet at 90 angles. Why do you need to say “in the same plane” for parallel lines but not for perpendicular lines? Sketch or demonstrate a counterexample to show the following definition is incomplete: “Parallel lines are lines that never meet.” (Two lines that do not intersect and are noncoplanar are skew lines.) 48 CHAPTER 1 Introducing Geometry Investigation Defining Angles Here are some examples and non-examples of special types of angles. Step 1 Step 2 Step 3 Step 4 Write a definition for each boldfaced term. Make sure your definitions highlight important differences. Trade definitions and test each other’s definitions by looking for counterexamples. If another group member finds a countere |
xample to one of your definitions, write a better definition. As a group, decide on the best definition for each term. As a class, agree on common definitions. Add these to your notebook. Draw and label a picture to illustrate each definition. Right Angle 90° 46° 105° Right angles Not right angles Acute Angle 89° 33° 91° 56° 100° Acute angles Not acute angles Notice the many congruent angles in this Navajo transitional Wedgeweave blanket. Are they right, acute, or obtuse angles? Obtuse Angle 101° 91° 129° 38° 89° Obtuse angles Not obtuse angles LESSON 1.3 What’s a Widget? 49 Pair of Vertical Angles 10 Pairs of vertical angles: Not pairs of vertical angles: 1 and 2 3 and 4 AED and BEC AEC and DEB Linear Pair of Angles C A E B D 1 2 3 4 Linear pairs of angles: 1 and 2 3 and 4 AED and AEC BED and DEA Pair of Complementary Angles m1 m2 90° 1 2 3 4 1 and 2 3 and 4 5 and6 7 and 8 9 and 10 1 2 30° 3 4 A B 150° 5 6 Not linear pairs of angles: 1 and 2 3 and 4 5 and 6 A and B m1 m2 90° G 40° 52° H 3 4 1 2 Pairs of complementary angles: 1 and 2 3 and 4 Not pairs of complementary angles: G and H 1 and 2 3 and 4 Pair of Supplementary Angles m3 m4 180° m4 m5 180 Pairs of supplementary angles: Not pairs of supplementary angles: 1 and 2 3 and 4 1, 2, and 3 4 and 5 What types of angles or angle pairs do you see in this magnified view of a computer chip? 50 CHAPTER 1 Introducing Geometry How did you do? Did you notice the difference between a supplementary pair and a linear pair? Did you make it clear which is which in your definitions? The more you practice writing geometry definitions, the better you will get at it. The design of this African Kente cloth contains examples of parallel and perpendicular lines, obtuse and acute angles, and complementary and supplementary angle pairs. To learn about the significance of Kente cloth designs, visit www.keymath.com/DG. EXERCISES For Exercises 1–8, draw and carefully label the figures |
. Use the appropriate marks to indicate right angles, parallel lines, congruent segments, and congruent angles. Use a protractor and a ruler when you need to. 1. Acute angle DOG with a measure of 45° 3. Obtuse angle BIG with angle bisector IE 5. PE AR 7. Complementary angles A and B with mA 40° 2. Right angle RTE 4. DG MS 6. Vertical angles ABC and DBE 8. Supplementary angles C and D with mD 40° 9. Which creatures in the last group below are Zoids? What makes a Zoid a Zoid? A C B D Zoids Not Zoids Who are Zoids? 10. What are the characteristics of a good definition? For Exercises 11–20, four of the statements are true. Make a sketch or demonstrate each true statement. For each false statement, draw a counterexample. 11. For every line segment there is exactly one midpoint. 12. For every angle there is exactly one angle bisector. 13. If two different lines intersect, then they intersect at one and only one point. 14. If two different circles intersect, then they intersect at one and only one point. LESSON 1.3 What’s a Widget? 51 15. Through a given point on a line there is one and only one line perpendicular to the given line. 16. In every triangle there is exactly one right angle. 17. Through a point not on a line, one and only one line can be constructed parallel to the given line. 18. If CA AT, then A is the midpoint of CT. 19. If mD 40° and mC 140°, then angles C and D are a linear pair. 20. If point A is not the midpoint of CT, then CA AT. 21. There is something wrong with this definition for a pair of vertical angles: “If AB and CD intersect at point P, then APC and BPD are a pair of vertical angles.” Sketch a counterexample to show why it is not correct. Can you add a phrase to correct it? y 5 S –5 T x 5 –5 R A B Mirror Review For Exercises 22 and 23, refer to the graph at right. 22. Find possible coordinates of a point P so that points P, T, and S are collinear. 23. Find possible coordinates of a point Q so that |
QR TS. 24. A partial mirror reflects some light and lets the rest of the light pass through. In the figure at right, half the light from point A passes through the partial mirror to point B. Copy the figure, then draw the outgoing angle for the light reflected from the mirror. What do you notice about the ray of reflected light and the ray of light that passes through? Science Albert Abraham Michelson (1852–1931) designed the Michelson Interferometer to find the wavelength of light. A modern version of the experiment uses a partial mirror to split a laser beam so that it travels in two different directions, and mirrors to recombine the separated beams. Partial mirror Mirrors Lens Laser 52 CHAPTER 1 Introducing Geometry 25. Find possible coordinates of points A, B, and C so that BAC is a right angle, BAT is an acute angle, ABS is an obtuse angle, and the points C, T, and R are collinear. 26. If D is the midpoint of AC and C is the midpoint of AB, and AD 3 cm, what is the length of AB? 27. If BD is the angle bisector of ABC, BE is the angle bisector of ABD, and mDBC 24°, what is mEBC? 28. Draw and label a figure that has two congruent segments and three congruent angles. Mark the congruent angles and congruent segments. y 6 S –12 –6 x T R 29. Show how three lines in a plane can intersect in no points, exactly one point, exactly two points, or exactly three points. 30. Show how it is possible for two triangles to intersect in one point, two points, three points, four points, five points, or six points, but not seven points. Show how they can intersect in infinitely many points. 31. Each pizza is cut into slices from the center. a. What fraction of the pizza is left? b. What fraction of the pizza is missing? c. If the pizza is cut into nine equal slices, how many degrees is each angle at the center of the pizza? 60° 120° IMPROVING YOUR VISUAL THINKING SKILLS Polyominoes In 1953, United States mathematician Solomon Golomb introduced polyominoes at the Harvard Mathematics Club, and they have been played with and enjoyed throughout the world ever since. Polyominoes are shapes made by connecting congruent squares. The |
squares are joined together side to side. (A complete side must touch a complete side.) Some of the smaller polyominoes are shown below. There is only one monomino and only one domino, but there are two trominoes, as shown. There are five tetrominoes— one is shown. Sketch the other four. Monomino Domino Trominoes Tetromino LESSON 1.3 What’s a Widget? 53 L E S S O N 1.4 There are two kinds of people in this world: those who divide everything into two groups, and those who don’t. KENNETH BOULDING Polygons A polygon is a closed figure in a plane, formed by connecting line segments endpoint to endpoint with each segment intersecting exactly two others. Each line segment is called a side of the polygon. Each endpoint where the sides meet is called a vertex of the polygon. Polygons Not polygons You classify a polygon by the number of sides it has. Familiar polygons have specific names, listed in this table. The ones without specific names are called n-sided polygons, or n-gons. For instance, you call a 25-sided polygon a 25-gon. Consecutive angles E Consecutive vertices A B D C Consecutive sides To name a polygon, list the vertices in consecutive order. You can name the pentagon above pentagon ABCDE. You can also call it DCBAE, but not BCAED. When the polygon is a triangle, you use the triangle symbol. For example, ABC means triangle ABC. A C E D A B B C Pentagon ABCDE ABC A diagonal of a polygon is a line segment that connects two nonconsecutive vertices. A polygon is convex if no diagonal is outside the polygon. A polygon is concave if at least one diagonal is outside the polygon. Sides Name 3 4 5 6 7 8 9 10 11 12 n Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon Undecagon Dodecagon n-gon Diagonal E D F G Convex polygons: All diagonals are inside Concave polygons: One or more diagonals are outside 54 CHAPTER 1 Introducing Geometry Recall that two segments or two angles are congruent if and only if they have the same measures. Two |
polygons are congruent polygons if and only if they are exactly the same size and shape. “If and only if ” means that the statements work both ways. If polygons are congruent, then corresponding sides and angles are congruent. If corresponding sides and angles are congruent, then polygons are congruent. For example, if quadrilateral CAMP is congruent to quadrilateral SITE, then their four pairs of corresponding angles and four pairs of corresponding sides are also congruent. When you write a statement of congruence, always write the letters of the corresponding vertices in an order that shows the correspondences. How does the shape of the framework of this Marc Chagall (1887–1985) stained glass window support the various shapes of the design? P M E T C I A S CAMP SITE EXAMPLE Which polygon is congruent to ABCDE? ABCDE? F A 100 130 130 E D B C K J Q 100 T P 130 L 130 S U G H N M W V Solution All corresponding sides and angles must be congruent, so polygon ABCDE polygon QLMNP. You could also say ABCDE QPNML, because all the congruent parts would still match. In an equilateral polygon, all the sides have equal length. In an equiangular polygon, all the angles have equal measure. A regular polygon is both equilateral and equiangular. Equiangular octagon Equilateral octagon Regular octagon LESSON 1.4 Polygons 55 EXERCISES For Exercises 1–8, classify each polygon. Assume that the sides of the chips and crackers are straight. 1. 5. 2. 6. 3. 7. 4. 8. For Exercises 9–11, draw an example of each polygon. 9. Quadrilateral 10. Dodecagon 11. Octagon For Exercises 12 and 13, give one possible name for each polygon. 12. R 13. F O 14. B F I E V R U C 15. Name a pair of consecutive angles and a pair of consecutive sides in the L O figure below. H A C Y T N 16. Draw a concave hexagon. How many diagonals does it have? The repeating pattern of squares and triangles creates a geometric tree in this quilt design by Diane Venters. What other |
polygons can you find in this quilt? 56 CHAPTER 1 Introducing Geometry 17. Name the diagonals of pentagon ABCDE. For Exercises 18 and 19, use the information given to name the triangle that is congruent to the first one. D 18. EAR? A I 19. OLD? E R T N 20. In the figure at right, THINK POWER. a. Find the missing measures. b. If mP 87° and mW 165°, which angles in THINK do you know? Write their measures. 21. If pentagon FIVER is congruent to pentagon PANCH 58 N 20 I L K 34 T 28 44 H then which side in pentagon FIVER is congruent to side PA? Which angle in pentagon PANCH is congruent to IVE? 22. Draw an equilateral concave pentagon. Then draw an equiangular concave pentagon. For Exercises 23–26, copy the given polygon and segment onto graph paper. Give the coordinates of the missing points. 23. CAR PET y 24. TUNA FISH y 6 R P C –5 x 5 E A –6 25. BLUE FISH y 6 E U L 6 x –6 H F B –6 6 A H F N x 6 –7 T U –6 26. RECT ANGL y R L 7 G –7 E T x 12 C LESSON 1.4 Polygons 57 y C (0, 5) B (4, 4) x A (0, 5) 27. Draw an equilateral octagon ABCDEFGH with A(5, 0), B(4,4), and C(0, 5) as three of its vertices. Is it regular? For Exercises 28–32, sketch and carefully label the figure. Mark the congruences. 28. Pentagon PENTA with PE EN 29. Hexagon NGAXEH with HEX EXA 30. Equiangular quadrilateral QUAD with QU QD 31. A hexagon with exactly one line of reflectional symmetry 32. Two different equilateral pentagons with perimeter 25 cm 33. Use your compass, protractor, and straightedge to draw a regular pentagon. 34. A rectangle with perimeter 198 cm is divided into five congruent rectangles as shown in the diagram at right. What is the perimeter of one of the five congruent |
rectangles? Review 35. Name a pair of complementary angles and a pair of vertical angles in the figure at right. 36. Draw AB, CD, and EF with AB CD and CD EF. 37. Draw a counterexample to show that this statement is false: “If a rectangle has perimeter 50 meters, then a pair of adjacent sides measures 10 meters and 15 meters.” A S 18° T 72° O R C E 38. Is it possible for four lines in a plane to intersect in exactly zero points? One point? Two points? Three points? Four points? Five points? Six points? Draw a figure to support each of your answers. IMPROVING YOUR VISUAL THINKING SKILLS Pentominoes I In Polyominoes I, you learned about shapes called polyominoes. Polyominoes with five squares are called pentominoes. There are 12 pentominoes. Can you find them all? One is shown at right. Use graph paper or square dot paper to sketch all 12. 58 CHAPTER 1 Introducing Geometry L E S S O N 1.5 The difference between the right word and the almost right word is the difference between lightning and the lightning bug. MARK TWAIN Triangles and Special Quadrilaterals You have learned to be careful with geometry definitions. It turns out that you also have to be careful with diagrams. When you look at a diagram, be careful not to assume too much from it. To assume something is to accept it as true without facts or proof. Things you can assume: You may assume that lines are straight, and if two lines intersect, they intersect at one point. Lightning You may assume that points on a line are collinear and that all points shown in a diagram are coplanar unless planes are drawn to show that they are noncoplanar. Not lightning Things you can’t assume: You may not assume that just because two lines or segments look parallel that they are parallel—they must be marked parallel! You may not assume that two lines are perpendicular just because they look perpendicular—they must be marked perpendicular! Pairs of angles, segments, or polygons are not necessarily congruent, unless they are marked with information that tells you they must be congruent! EXAMPLE In the diagrams below, which pairs of lines are perpendicular? Which pairs of lines are parallel? Which pair of triangles is congruent LESSON 1.5 Tri |
angles and Special Quadrilaterals 59 Solution By studying the markings, you can tell that AB CD, JK JM, and STU XYZ. In this lesson you will write definitions that classify different kinds of triangles and special quadrilaterals, based on relationships among their sides and angles. Investigation Triangles and Special Quadrilaterals Write a good definition of each boldfaced term. Discuss your definitions with others in your group. Agree on a common set of definitions for your class and add them to your definition list. In your notebook, draw and label a figure to illustrate each definition. Right Triangle 75° 90° 26° 64° 15° 58° 87° 65° 91° 35° 24° Right triangles Not right triangles Acute Triangle 51° 63° 66° 40° 28° 80° 72° 85° 55° Acute triangles Obtuse Triangle 41° 112° 27° 118° 40° 22° 50° 27° 122° 31° 40° Not acute triangles 104° 130° 72° 68° 57° 88° 66° 26° 33° Obtuse triangles Not obtuse triangles What shape is the basis for the design on this Islamic textile from Uzbekistan? 60 CHAPTER 1 Introducing Geometry The Sol LeWitt (b 1928, United States) design inside this art museum uses triangles and quadrilaterals to create a painting the size of an entire room. Sol LeWitt, Wall Drawing #652— On three walls, continuous forms with color ink washes superimposed, color in wash. Collection: Indianapolis Museum of Art, Indianapolis, IN. September, 1990. Courtesy of the artist. Scalene Triangle 15 17 8 6 8 7 60° 28 20 11 6 5 11 8 4 Scalene triangles Not scalene triangles Equilateral Triangle 8 8 8 Equilateral triangles Not equilateral triangles Isosceles Triangle Isosceles triangles Not isosceles triangles LESSON 1.5 Triangles and Special Quadrilaterals 61 In an isosceles triangle, the angle between the two sides of equal length is called the vertex angle. The side opposite the vertex angle is called the base of the isoceles triangle. The two angles opposite the two sides of equal length are called the base angles of the isoceles triangle. Base Base angles Vertex angle Now let’s write definitions for quadrilaterals. Trapezoid Trapezoids Not trapezoids How does this ancient pyramid in Mexico use shapes for practical purposes and |
also for its overall attractiveness? At the Acoma Pueblo Dwellings in New Mexico, how do sunlight and shadows enhance the existing shape formations? How many shapes make up the overall triangular shapes of these pyramids at the Louvre in Paris? 62 CHAPTER 1 Introducing Geometry Kite Kites Not kites Recreation Today’s kite designers use lightweight plastics, synthetic fabrics, and complex shapes to sustain kites in the air longer than earlier kites that were made of wood, cloth, and had the basic “kite” shape. Many countries even hold annual kite festivals where contestants fly flat kites, box kites, and fighter kites. The design will determine the fastest and most durable kite in the festival. Parallelogram Rhombus Rectangle Parallelograms Not parallelograms Rhombuses Not rhombuses 90° 90° 90° 90° Rectangles Not rectangles LESSON 1.5 Triangles and Special Quadrilaterals 63 Square S P D R Q A C B Squares Not squares As you learned in the investigation, a square is not a square unless it has the proper markings. Keep this in mind as you work on the exercises. EXERCISES 1. Based on the marks, what can you assume to be true in each figure For Exercises 2–9, match the term on the left with its figure on the right. A. E. G. 2. Scalene right triangle 3. Isosceles right triangle 4. Isosceles obtuse triangle 5. Trapezoid 6. Rhombus 7. Rectangle 8. Kite 9. Parallelogram B. F. C. D. H. I. For Exercises 10–18, sketch and label the figure. Mark the figures. 10. Isosceles acute triangle ACT with AC CT 11. Scalene triangle SCL with angle bisector CM 12. Isosceles right triangle CAR with mCRA 90° 64 CHAPTER 1 Introducing Geometry 13. Trapezoid ZOID with ZO ID 14. Kite BENF with BE EN 15. Rhombus EQUL with diagonals EU and QL intersecting at A 16. Rectangle RGHT with diagonals RH and GT intersecting at I 17. Two different isosceles triangles with perimeter 4a b 18. Two noncongruent triangles, each with side 6 cm and an |
angle measuring 40° 19. Draw a hexagon with exactly two outside diagonals. 20. Draw a regular quadrilateral. What is another name for this shape? 21. Find the other two vertices of a square with one vertex (0, 0) and another vertex (4, 2). Can you find another answer? Austrian architect and artist Friedensreich Hundertwasser (1928–2000) designed the apartment house Hundertwasser-House (1986) with a square spiral staircase in its center. What other shapes do you see? For Exercises 22–24, use the graphs below. Can you find more than one answer? y 5 x –5 –5 R Y –5 y 5 –5 M –5 E y 5 C –5 x –5 L x –5 –5 22. Locate a point L so that LRY is an isosceles triangle. 23. Locate a point O so that MOE is an isosceles right triangle. 24. Locate a point R so that CRL is an isosceles right triangle. Review For Exercises 25–29, tell whether the statement is true or false. For each false statement, sketch a counterexample or explain why the statement is false. 25. An acute angle is an angle whose measure is less than 90°. 26. If two lines intersect to form a right angle, then the lines are perpendicular. 27. A diagonal is a line segment that connects any two vertices of a polygon. LESSON 1.5 Triangles and Special Quadrilaterals 65 28. A ray that divides the angle into two angles is the angle bisector. y 29. An obtuse triangle has exactly one angle whose measure is greater than 90°. (–3, 5) (2, 4) 30. Use the ordered pair rule (x, y) → (x 1, y 3) to relocate the (–4, 1) four vertices of the given quadrilateral. Connect the four new points to create a new quadrilateral. Do the two quadrilaterals appear congruent? Check your guess with tracing paper or patty paper. (1, 1) x 31. Suppose a set of thin rods is glued together into a triangle as shown. How would you place the triangular arrangement of rods onto the edge of a ruler so that they balance? Explain why. DRAWING THE IMPOSSIBLE You experienced |
some optical illusions with op art in Chapter 0. Some optical illusions are tricks—they at first appear to be drawings of real objects, but actually they are impossible to make, except on paper. Reproduce the two impossible objects by drawing them on full sheets of paper. Can you create an impossible object of your own? Try it. From WALTER WICK’S OPTICAL TRICKS. Published by Cartwheel Books, a division of Scholastic Inc. ©1998 by Walter Wick. Reprinted by permission. 66 CHAPTER 1 Introducing Geometry L E S S O N 1.6 I can never remember things I didn’t understand in the first place. AMY TAN Circles Unless you walked to school this morning, you arrived on a vehicle with circular wheels. A circle is the set of all points in a plane at a given distance (radius) from a given point (center) in the plane. You name a circle by its center. The circle on the bicycle wheel, with center O, is called circle O. When you see a dot at the center of a circle, you can assume that it represents the center point. A segment from the center to a point on the edge of the circle is called a radius. Its length is also called the radius. The diameter is a line segment containing the center, with its endpoints on the circle. The length of this segment is also called the diameter. By permission of Johnny Hart and Creators Syndicate, Inc. O LESSON 1.6 Circles 67 If two or more circles have the same radius, they are congruent circles. If two or more coplanar circles share the same center, they are concentric circles. Congruent circles Concentric circles An arc of a circle is two points on the circle and the continuous (unbroken) part of the circle between the two points. The two points are called the endpoints of the arc. A B or BA. You classify arcs into three types: semicircles, minor You write arc AB as AB arcs, and major arcs. A semicircle is an arc of a circle whose endpoints are the endpoints of a diameter. A minor arc is an arc of a circle that is smaller than a semicircle. A major arc is an arc of a circle that is larger than a semicircle. You can name minor arcs with the letters of the two endpoints. For semicircles and major arcs, you need three points |
to make clear which arc you mean—the first and last letters are the endpoints and the middle letter is any other point on the arc. P Minor arc AP Semicircle APD A O D Major arc PAD Arcs have a degree measure, just as angles do. A full circle has an arc measure of 360°, a semicircle has an arc measure of 180°, and so on. You find the arc measure by measuring the central angle, the angle with its vertex at the center of the circle, and sides passing through the endpoints of the arc. BC a minor arc mBC 45° B 45° C BDC a major arc mBDC 315° 315° D 68 CHAPTER 1 Introducing Geometry Investigation Defining Circle Terms Step 1 Write a good definition of each boldfaced term. Discuss your definitions with others in your group. Agree on a common set of definitions as a class and add them to your definition list. In your notebook, draw and label a figure to illustrate each definition. Chord Chords: AB, CD, EF, GH, and IJ Diameter C A E O F B D Not chords: PQ, RS, TU, and VW R T W P S Q V U Diameters: AB, CD, and EF Not diameters: PQ, RS, TU, and VW Tangent Tangents: AB, CD, and EF Not tangents: PQ, RS, TU, and VW Note: You can say AB is a tangent, or you can say AB is tangent to circle O. The point where the tangent touches the circle is called the point of tangency. Step 2 Step 3 Can a chord of a circle also be a diameter of the circle? Can it be a tangent? Explain why or why not. Can two circles be tangent to the same line at the same point? Draw a sketch and explain. LESSON 1.6 Circles 69 In each photo, find examples of the terms introduced in this lesson. Circular irrigation on a farm Japanese wood bridge CAD design of pistons in a car engine EXERCISES For Exercises 1–8, use the diagram at right. Points E, P, and C are collinear. 1. Name three chords. 2. Name one diameter. 3. Name five radii. 4. Name five minor arcs. 5. Name two semicircles. 6. Name two major arcs. |
7. Name two tangents. 8. Name a point of tangency. 9. Name two types of vehicles that use wheels, two household appliances that use wheels, and two uses of the wheel in the world of entertainment. 10. In the figure at right, what is mPQ?? mPRQ 11. Use your compass and protractor to make an arc with measure 65°. Now make an arc with measure 215°. Label each arc with its measure 110° O R 70 CHAPTER 1 Introducing Geometry 12. Name two places or objects where concentric circles appear. Bring an example of a set of concentric circles to class tomorrow. You might look in a magazine for a photo or make a copy of a photo from a book (but not this book!). 13. Sketch two circles that appear to be concentric. Then use your compass to construct a pair of concentric circles. 14. Sketch circle P. Sketch a triangle inside circle P so that the three sides of the triangle are chords of the circle. This triangle is “inscribed” in the circle. Sketch another circle and label it Q. Sketch a triangle in the exterior of circle Q so that the three sides of the triangle are tangents of the circle. This triangle is “circumscribed” about the circle. 15. Use your compass to construct two circles with the same radius intersecting at two points. Label the centers P and Q. Label the points of intersection of the two circles A and B. Construct quadrilateral PAQB. What type of quadrilateral is it? 16. Do you remember the daisy construction from Chapter 0? Construct a circle with radius s. With the same compass setting, divide the circle into six congruent arcs. Construct the chords to form a regular hexagon inscribed in the circle. Construct radii to each of the six vertices. What type of triangles are formed? What is the ratio of the perimeter of the hexagon to the diameter of the circle? 17. Sketch the path made by the midpoint of a radius of a circle if the radius is rotated about the center. For Exercises 18–20, use the ordered pair rule shown to relocate the four points on the given circle. Can the four new points be connected to create a new circle? Does the new figure appear congruent to the original circle? 18. (x, y) → (x 1, y 2) y 19. (x, y) → ( |
2x, 2y) 20. (x, y) → (2x, y5 x 5 –5 5 –5 –5 Review For Exercises 21–24, draw each kind of triangle or write “not possible” and explain why. Use your geometry tools to make your drawings as accurate as possible. 21. Isosceles right triangle 23. Scalene obtuse triangle 22. Scalene isosceles triangle 24. Isosceles obtuse triangle LESSON 1.6 Circles 71 For Exercises 25–33, sketch, label, and mark the figure. 25. Obtuse scalene triangle FAT with mFAT 100° 26. Trapezoid TRAP with TR AP and TRA a right angle 27. Two different (noncongruent) quadrilaterals with angles of 60°, 60°, 120°, and 120° 28. Equilateral right triangle 29. Right isosceles triangle RGT with RT GT and mRTG 90° 30. An equilateral triangle with perimeter 12a 6b 31. Two triangles that are not congruent, each with angles measuring 50° and 70° 32. Rhombus EQUI with perimeter 8p and mIEQ 55° 33. Kite KITE with TE 2EK and mTEK 120° IMPROVING YOUR REASONING SKILLS Checkerboard Puzzle 1. Four checkers—three red and one black—are arranged on the corner of a checkerboard, as shown at right. Any checker can jump any other checker. The checker that was jumped over is then removed. With exactly three horizontal or vertical jumps, remove all three red checkers, leaving the single black checker. Record your solution. 2. Now, with exactly seven horizontal or vertical jumps, remove all seven red checkers, leaving the single black checker. Record your solution 72 CHAPTER 1 Introducing Geometry L E S S O N 1.7 You can observe a lot just by watching. YOGI BERRA A Picture Is Worth a Thousand Words A picture is worth a thousand words! That expression certainly applies to geometry. A drawing of an object often conveys information more quickly than a long written description. People in many occupations use drawings and sketches to communicate ideas. Architects create blueprints. Composers create musical scores. Choreographers visualize and map out sequences of dance steps. Basketball coaches design plays. Interior designers— |
well, you get the picture. Visualization skills are extremely important in geometry. So far, you have visualized geometric situations in every lesson. To visualize a plane, you pictured a flat surface extending infinitely. In another lesson, you visualized the number of different ways that four lines can intersect. Can you picture what the hands of a clock look like when it is 3:30? 10 9 8 12 1 11 By drawing diagrams, you apply visual thinking to problem solving. Let’s look at some examples that show how to use visual thinking to solve word problems. EXAMPLE A Volumes 1 and 2 of a two-volume set of math books sit next to each other on a shelf. They sit in their proper order: Volume 1 on the left and Volume 2 on the right. Each front and back cover is 1 inch thick, and the 8 pages portion of each book is 1 inch thick. If a bookworm starts at the first page of Volume 1 and burrows all the way through to the last page of Volume 2, how far will she travel? Take a moment and try to solve the problem in your head. Bookplate for Albert Ernst Bosman, M. C. Escher, 1946 ©2002 Cordon Art B. V.–Baarn–Holland. All rights reserved. LESSON 1.7 A Picture Is Worth a Thousand Words 73 Solution Did you get 21 inches? It seems reasonable, doesn’t it? 4 Guess what? That’s not the answer. Let’s get organized. Reread the problem to identify what information you are given and what you are trying to find. First page of Volume 1 You are given the thickness of each cover, the thickness of the page portion, and the position of the books on the shelf. You are trying to find how far it is from the first page of Volume 1 to the last page of Volume 2. Draw a picture and locate the position of the pages referred to in the problem. Now “look” how easy it is to solve the problem. She traveled only 1 inch through the two covers! 4 Last page of Volume 2 EXAMPLE B In Reasonville, many streets are named after famous mathematicians. Streets that end in an “s” run east–west. All other streets might run either way. Wiles Street runs perpendicular to Germain Street. Fermat Street runs parallel to Germain Street. Which direction does Fermat Street run? Mathematics Sophie Germain |
(1776–1831), a French mathematician with no formal education, wrote a prize treatise, contributed to many theories, and worked extensively on Fermat’s Last Theorem. The French mathematician Pierre de Fermat (1601–1665) developed analytic geometry. His algebraic approach is what made his influence on geometry so strong. Andrew Wiles (b 1953), an English mathematician at Princeton University, began trying to prove Fermat’s Last Theorem when he was just 10 years old. In 1993, after spending most of his career working on the theorem, sometimes in complete isolation, he announced a proof of the problem. Solution Did you make a diagram? You can start your diagram with the first piece of information. Then you can add to the diagram as new information is added. Wiles Street ends in an “s,” so it runs east–west. You are trying to find the direction of Fermat Street. 74 CHAPTER 1 Introducing Geometry EXAMPLE C N N W Wiles E W Wiles E W Wiles E Germain Germain Fermat S S Initial diagram Improved diagram Final diagram The final diagram reveals the answer. Fermat Street runs north–south. Sometimes there is more than one point or even many points that satisfy a set of conditions. The set of points is called a locus of points. Let’s look at an example showing how to solve a locus problem. Harold, Dina, and Linda are standing on a flat, dry field reading their treasure map. Harold is standing at one of the features marked on the map, a gnarled tree stump, and Dina is standing atop a large black boulder. The map shows that the treasure is buried 60 meters from the tree stump and 40 meters from the large black boulder. Harold and Dina are standing 80 meters apart. What is the locus of points where the treasure might be buried? Solution Start by drawing a diagram based on the information given in the first two sentences, then add to the diagram as new information is added. Can you visualize all the points that are 60 meters from the tree stump? Mark them on your diagram. They should lie on a circle. The treasure is also 40 meters from the boulder. All the possible points lie in a circle around the boulder. The two possible spots where the treasure might be buried, or the locus of points, are the points where the two circles intersect. H 80 m D 60 m H D 60 m H |
Treasure here 40 m D Or here Initial diagram Improved diagram Final diagram LESSON 1.7 A Picture Is Worth a Thousand Words 75 EXERCISES 1. Surgeons, engineers, carpenters, plumbers, electricians, and furniture movers all rely on trained experience with visual thinking. Describe how one of these tradespeople or someone in another occupation uses visual thinking in his or her work. You will need Construction tools for Exercises 33 and 34 Now try your hand at some word problems. Read each problem carefully, determine what you are trying to find, and draw and label a diagram. Finally, solve the problem. 2. In the city of Rectangulus, all the streets running east–west are numbered and those streets running north–south are lettered. The even-numbered streets are one-way east and the odd-numbered streets are one-way west. All the vowel-lettered avenues are one-way north and the rest are two-way. Can a car traveling south on S Street make a legal left turn onto 14th Street? 3. Freddie the Frog is at the bottom of a 30-foot well. Each day he jumps up 3 feet, but then, during the night, he slides back down 2 feet. How many days will it take Freddie to get to the top and out? 4. Mary Ann is building a fence around the outer edge of a rectangular garden plot that measures 25 feet by 45 feet. She will set the posts 5 feet apart. How many posts will she need? 5. Midway through a 2000-meter race, a photo is taken of five runners. It shows Meg 20 meters behind Edith. Edith is 50 meters ahead of Wanda, who is 20 meters behind Olivia. Olivia is 40 meters behind Nadine. Who is ahead? In your diagram, use M for Meg, E for Edith, and so on. 6. Here is an exercise taken from Marilyn vos Savant’s Ask Marilyn® column in Parade magazine. It is a good example of a difficultsounding problem becoming clear once a diagram has been made. Try it. A 30-foot cable is suspended between the tops of two 20-foot poles on level ground. The lowest point of the cable is 5 feet above the ground. What is the distance between the two poles? 7. Points A and B lie in a plane. Sketch the locus of points in the plane that are equally distant from points A and B. Sketch the |
locus of points in space that are equally distant from points A and B. 8. Draw an angle. Label it A. Sketch the locus of points in the plane of angle A that are the same distance from the two sides of angle A. 9. Line AB lies in plane. Sketch the locus of points in plane that are 3 cm from AB. Sketch the locus of points in space that are 3 cm from AB. 76 CHAPTER 1 Introducing Geometry 10. Beth Mack and her dog Trouble are exploring in the woods east of Birnam Woods Road, which runs north-south. They begin walking in a zigzag pattern: 1 km south, 1 km west, 1 km south, 2 km west, 1 km south, 3 km west, and so on. They walk at the rate of 4 km/h. If they started 15 km east of Birnam Woods Road at 3:00 P.M., and the sun sets at 7:30 P.M., will they reach Birnam Woods Road before sunset? In geometry you will use visual thinking all the time. In Exercises 11 and 12 you will be asked to locate and recognize congruent geometric figures even if they are in different positions due to translations (slides), rotations (turns), or reflections (flips). 11. If trapezoid ABCD were rotated 90° counterclockwise about (0, 0), to what (x, y) location would points A, B, C, and D be relocated? 12. If CYN were reflected over the y-axis, to what location would points C, N, and Y be relocated? y D (1, 2) C (3, 2) A (0, 0) x B (5, 0) y N (0, 3) Y (4, 1) x C (–3, –1) 13. What was the ordered pair rule used to relocate the four vertices of ABCD to ABCD4 A A –4 14. Which lines are perpendicular? Which lines are parallel 15. Sketch the next two figures in the pattern below. If this pattern were to continue, what would be the perimeter of the eighth figure in the pattern? (Assume the length of each segment is 1 cm.),,,..., LESSON 1.7 A Picture Is Worth a Thousand Words 77 16. Many of the geometric figures you have defined are closely related to one another. A diagram can help you |
see the relationships among them. At right is a concept map showing the relationships among members of the triangle family. This type of concept map is known as a tree diagram because the relationships are shown as branches of a tree. Copy and fill in the missing branches of the tree diagram for triangles. 17. At right is a concept map showing the relationships among some members of the parallelogram family. This type of concept map is known as a Venn diagram. Fill in the missing names. Acute Triangles Obtuse?? Scalene Isosceles Isosceles Isosceles Scalene Equilateral Parallelograms? Squares? A net is a two-dimensional pattern that you can cut and fold to form a three-dimensional figure. Another visual thinking skill you will need is the ability to visualize nets being folded into solid objects and geometric solids being unfolded into nets. The net below left can be folded into a cube and the net below right can be folded into a pyramid. Net for a cube Net for a square-based pyramid 18. Which net(s) will fold to make a cube? A. D. B. E. C. F. For Exercises 19–22, match the net with its geometric solid. 19. A. 20. B. 21. C. 22. D. 78 CHAPTER 1 Introducing Geometry Review For Exercises 23–32, write the words or the symbols that make the statement true. 23. The three undefined terms of geometry are?,?, and?. 24. “Line AB” may be written using a symbol as?. 25. “Arc AB” may be written using a symbol as?. 26. The point where the two sides of an angle meet is the? of the angle. 27. “Ray AB” may be written using a symbol as?. 28. “Line AB is parallel to segment CD” is written in symbolic form as?. 29. The geometry tool you use to measure an angle is a?. 30. “Angle ABC” is written in symbolic form as?. 31. The sentence “Segment AB is perpendicular to line CD” is written in symbolic form as?. 32. The angle formed by a light ray coming into a mirror is? the angle formed by a light ray leaving the mirror. William Thomas Williams, DO YOU THINK A IS B, acrylic on canvas, 1975–77, |
Fisk University Galleries, Nashville, Tennessee. 33. Use your compass to draw two congruent circles intersecting in exactly one point. How does the distance between the two centers compare with the radius? 34. Use your compass to construct two congruent circles so that each circle passes through the center of the other circle. Label the centers P and Q. Construct PQ connecting the centers. Label the points of intersection of the two circles A and B. Construct chord AB. What is the relationship between AB and PQ? IMPROVING YOUR VISUAL THINKING SKILLS Hexominoes Polyominoes with six squares are called hexominoes. There are 35 different hexominoes. There is 1 with a longest string of six squares; there are 3 with a longest string of five squares, 13 with a longest string of four squares, 17 with a longest string of three squares; and there is 1 with a longest string of two squares. Use graph paper to sketch the 35 hexominoes which are nets for a cube. Here is one hexomino that does fold into a cube. LESSON 1.7 A Picture Is Worth a Thousand Words 79 L E S S O N 1.8 When curiosity turns to serious matters, it’s called research. MARIE VON EBNERESCHENBACH Space Geometry Lesson 1.1 introduced you to point, line, and plane. Throughout this chapter you have used these terms to define a wide range of other geometric figures, from rays to polygons. You did most of your work on a single flat surface, a single plane. Some problems, however, required you to step out of a single plane to visualize geometry in space. In this lesson you will learn more about space geometry, or solid geometry. Space is the set of all points. Unlike lines and planes, space cannot be contained in a flat surface. Space is three-dimensional, or “3-D.” In an “edge view,” you see the front edge of a building as a vertical line, and the other edges as diagonal lines. Isometric dot paper helps you draw these lines, as you can see in the steps below. Let’s practice the visual thinking skill of presenting three-dimensional (3-D) objects in two-dimensional (2-D) drawings. The geometric solid you are probably most familiar with is a box, or rectangular prism. Below are steps for making |
a two-dimensional drawing of a rectangular prism. This type of drawing is called an isometric drawing. It shows three sides of an object in one view (an edge view). This method works best with isometric dot grid paper. After practicing, you will be able to draw the box without the aid of the dot grid. Step 1 Step 2 Step 3 Step 4 Use dashed lines for edges that you couldn’t see if the object were solid. 80 CHAPTER 1 Introducing Geometry The three-dimensional objects you will study include the six types of geometric solids shown below. Prism Pyramid Cylinder Cone Sphere Hemisphere The shapes of these solids are probably already familiar to you even if you are not familiar with their proper names. The ability to draw these geometric solids is an important visual thinking skill. Here are some drawing tips. Remember to use dashes for the hidden lines. Step 1 Step 2 Step 3 Step 4 Step 1 Step 2 Step 1 Step 2 Step 3 Step 4 LESSON 1.8 Space Geometry 81 Pyramid Cylinder Prism Cone Sphere Step 1 Step 2 Step 1 Step 2 Hemisphere Step 1 Step 2 Solid geometry also involves visualizing points and lines in space. In the following investigation, you will have to visualize relationships between geometric figures in a plane and in space. Investigation Space Geometry Step 1 Step 2 Make a sketch or use physical objects to demonstrate each statement in the list below. Work with your group to determine whether each statement is true or false. If the statement is false, draw a picture and explain why it is false. 1. Only one line can be drawn through two different points. 2. Only one plane can pass through one line and a point that is not on the line. 3. If two coplanar lines are both perpendicular to a third line in the same plane, then the two lines are parallel. 4. If two planes do not intersect, then they are parallel. 5. If two lines do not intersect, then they must be parallel. 6. If a line is perpendicular to two lines in a plane, but the line is not contained in the plane, then the line is perpendicular to the plane. 82 CHAPTER 1 Introducing Geometry EXERCISES For Exercises 1–6, draw each figure. Study the drawing tips provided on the previous page before you start. 1. Cylinder 2. Cone 3. Prism with a hexagonal base 4. Sphere 5. Pyramid with a heptagonal base 6. Hemisphere |
7. The photo at right shows a prism-shaped building with a pyramid roof and a cylindrical porch. Draw a cylindrical building with a cone roof and a prismshaped porch. For Exercises 8 and 9, make a drawing to scale of each figure. Use isometric dot grid paper. Label each figure. (For example, in Exercise 8, draw the solid so that the dimensions measure 2 units by 3 units by 4 units, then label the figure with meters.) 8. A rectangular solid 2 m by 3 m by 4 m, sitting on its A police station, or koban, in Tokyo, Japan biggest face. 9. A rectangular solid 3 inches by 4 inches by 5 inches, resting on its smallest face. Draw lines on the three visible surfaces showing how you can divide the solid into cubic-inch boxes. How many such boxes will fit in the solid? For Exercises 10–12, use isometric dot grid paper to draw the figure shown. 10. 11. 12. For Exercises 13–15, sketch the three-dimensional figure formed by folding each net into a solid. Name the solid. 13. 14. 15. LESSON 1.8 Space Geometry 83 For Exercises 16 and 17, find the lengths x and y. (Every angle on each block is a right angle.) 16. 20 45 x 17 y 13 45 18 17. 3 8 y 2 2 7 x 2 In Exercises 18 and 19, each figure represents a two-dimensional figure with a wire attached. The three-dimensional solid formed by spinning the figure on the wire between your fingers is called a solid of revolution. Sketch the solid of revolution formed by each two-dimensional figure. 18. 19. When a solid is cut by a plane, the resulting two-dimensional figure is called a section. For Exercises 20 and 21, sketch the section formed when each solid is sliced by the plane, as shown. 20. 21. A real-life example of a “solid of revolution” is a clay pot on a potter’s wheel. Slicing a block of clay reveals a section of the solid. Here, the section is a rectangle. 84 CHAPTER 1 Introducing Geometry All of the statements in Exercises 22–29 are true except for two. Make a sketch to demonstrate each true statement. For each false statement, draw a sketch and explain why it is false. 22. Only one plane |
can pass through three noncollinear points. 23. If a line intersects a plane that does not contain the line, then the intersection is exactly one point. 24. If two lines are perpendicular to the same line, then they are parallel. 25. If two different planes intersect, then their intersection is a line. Physical models can help you visualize the intersections of lines and planes in space. Can you see examples of intersecting lines in this photo? Parallel lines? Planes? Points? 26. If a line and a plane have no points in common, then they are parallel. 27. If a plane intersects two parallel planes, then the lines of intersection are parallel. 28. If three random planes intersect (no two are parallel and all three do not share the same line), then they divide space into six parts. 29. If two lines are perpendicular to the same plane, then they are parallel to each other. Review 30. If the kite DIAN were rotated 90° clockwise about the origin, to what location would point A be relocated? 31. Use your ruler to measure the perimeter of WIM (in centimeters) and your protractor to measure the largest angle. 32. Use your geometry tools to draw a triangle with two sides of length 8 cm and length 13 cm and the angle between them measuring 120°. y N (0, 0) x D (–3, –1) I (–3, –3) A (–1, –3) I W M IMPROVING YOUR VISUAL THINKING SKILLS Equal Distances Here’s a real challenge: Show four points A, B, C, and D so that AB BC AC AD BD CD. LESSON 1.8 Space Geometry 85 Geometric Probability I You probably know what probability means. The probability, or likelihood, of a particular outcome is the ratio of the number of successful outcomes to the number of possible outcomes. So the probability of rolling a 4 on a 6-sided die is 1. Or, you can name 6 an event that involves more than one outcome, like getting the total 4 on two 6-sided dice. Since each die can come up in six different ways, there are 6 6 or 36 combinations (count ’em!). You can get the total 4 with a 1 and a 3, a 3 and a 1, or a 2 and a 2. So the probability 1 3 of getting the total 4 is. or 1 2 3 6 Anyway, that |
’s the theory. Activity Chances Are In this activity you’ll see that you can apply probability theory to geometric figures. The Spinner After you’ve finished your homework and have eaten dinner, you play a game of chance using the spinner at right. Where the spinner lands determines how you’ll spend the evening. Sector A: Playing with your younger brother the whole evening Sector B: Half the evening playing with your younger brother and half the evening watching TV Sector C: Cleaning the birdcage, the hamster cage, and the aquarium the whole evening Sector D: Playing in a band in a friend’s garage the whole evening You will need ● a protractor ● a ruler 86 CHAPTER 1 Introducing Geometry Step 1 Step 2 What is the probability of landing in each sector? What is the probability that you’ll spend at least half the evening with your younger brother? What is the probability that you won’t spend any time with him? The Bridge A computer programmer who is trying to win money on a TV survival program builds a 120 ft rope bridge across a piranha-infested river 90 ft below. Step 3 Step 4 If the rope breaks where he is standing (a random point), but he is able to cling to one end of it, what is the probability that he’ll avoid getting wet (or worse)? Suppose the probability that the rope breaks at all is 1. Also suppose that, as long 2 as he doesn’t fall more than 30 ft, the probability that he can climb back up is 3. 4 What is the probability that he won’t fall at all? What is the probability that if he does, he’ll be able to climb back up? The Bus Stop Noriko arrives at the bus stop at a random time each day. Her bus stops there every 20 minutes between 3:00 and 4:30 P.M. Step 5 Step 6 Draw a number line to show stopping times. (Don’t worry about the length of time that the bus is actually stopped. Assume it is 0 minutes.) What is the probability that she will have to wait 5 minutes or more? 10 minutes or more? Hint: What line lengths represent possible waiting time? Step 7 If the bus stops for exactly 3 minutes, how do your answers to Step 5 change? Step 8 Step 9 List the geometric properties you needed in each of the three scenarios above and tell how your answers depended |
on them. How is geometric probability like the probability you’ve studied before? How is it different? Step 10 Create your own geometric probability problem. EXPLORATION Geometric Probability 1 87 ● CHAPTER 11 REVIEW ● CHAPTER 1 REVIEW ● CHAPTER 1 REVIEW ● CHAPTER 1 REVIEW ● CHAPTER CHAPTER 1 R E V I E W It may seem that there’s a lot to memorize in this chapter. But having defined terms yourself, you’re more likely to remember and understand them. The key is to practice using these new terms and to be organized. Do the following exercises, then read Assessing What You’ve Learned for tips on staying organized. Whether you’ve been keeping a good list or not, go back now through each lesson in the chapter and double-check that you’ve completed each definition and that you understand it. For example, if someone mentions a geometry term to you, can you sketch it? If you are shown a geometric figure can you classify it? Compare your list of geometry terms with the lists of your group members. EXERCISES For Exercises 1–16, identify the statement as true or false. For each false statement, explain why it is false or sketch a counterexample. 1. The three basic building blocks of geometry are point, line, and plane. 2. “The ray through point P from point Q” is written in symbolic form as PQ. 3. “The length of segment PQ”can be written as PQ. 4. The vertex of angle PDQ is point P. 5. The symbol for perpendicular is. 6. A scalene triangle is a triangle with no two sides the same length. 7. An acute angle is an angle whose measure is more than 90°. 8. If AB intersects CD at point P, then APD and APC are a pair of vertical angles. 9. A diagonal is a line segment in a polygon connecting any two nonconsecutive vertices. 10. If two lines lie in the same plane and are perpendicular to the same line, then they are parallel. 11. If the sum of the measures of two angles is 180°, then the two angles are complementary. 12. A trapezoid is a quadrilateral having exactly one pair of parallel sides. 13. A polygon with ten sides is a decagon. 88 CHAPTER 1 Introducing Geometry |
A knowledge of parallel lines, planes, arcs, circles, and symmetry is necessary to build durable guitars that sound pleasing. ● CHAPTER 1 REVIEW ● CHAPTER 1 REVIEW ● CHAPTER 1 REVIEW ● CHAPTER 1 REVIEW ● CHAPTER 1 14. A square is a rectangle with all the sides equal in length. 15. A pentagon has five sides and six diagonals. 16. The largest chord of a circle is a diameter of the circle. For Exercises 17–25, match each term with its figure below, or write “no match.” 17. Isosceles acute triangle 18. Isosceles right triangle 19. Rhombus 20. Trapezoid 23. Concave polygon 21. Pyramid 24. Chord 22. Cylinder 25. Minor arc A. E. I.? B. F. J. M. N. D. H.? L. C. G. K. O. For Exercises 26–33, sketch, label, and mark each figure. 26. Kite KYTE with KY YT 27. Scalene triangle PTS with PS 3, ST 5, PT 7, and angle bisector SO 28. Hexagon REGINA with diagonal AG parallel to sides RE and NI 29. Trapezoid TRAP with AR and PT the nonparallel sides. Let E be the midpoint of PT and let Y be the midpoint of AR. Draw EY. 30. A triangle with exactly one line of reflectional symmetry 31. A circle with center at P, radii PA and PT, and chord TA creating a minor arc TA CHAPTER 1 REVIEW 89 EW ● CHAPTER 1 REVIEW ● CHAPTER 1 REVIEW ● CHAPTER 1 REVIEW ● CHAPTER 1 REVIEW ● CHAPTE 32. A pair of concentric circles with the diameter AB of the inner circle perpendicular at B to a chord CD of the larger circle 33. A pyramid with a pentagonal base 34. Draw a rectangular prism 2 inches by 3 inches by 5 inches, resting on its largest face. Draw lines on the three visible faces, showing how the solid can be divided into 30 smaller cubes. 35. Use your protractor to draw a 125° angle. 36. Use your protractor, ruler, and compass to draw an isosceles triangle with a vertex angle having a measure of 40°. 37. Use your geomety tools to draw a regular octagon. 38. What |
is the measure of A? Use your protractor. For Exercises 39–42, find the lengths x and y. y x 39. 42. 40. 1 3 8 3 2 4 4 18 y 7 x 2 2 2 20 A 41. y x x x 10 12 2y y 8 x y 12 43. If D is the midpoint of AC, C is the midpoint of AB, and BD 12 cm, what is the length of AB? 44. If BD is the angle bisector of ABC and BE is the angle bisector of DBC, find mEBA if mDBE 32°? 45. What is the measure of the angle formed by the hands of the clock at 2:30? 46. If the pizza is cut into 12 congruent pieces, how many degrees are in each central angle? 47. Make a concept map (a tree diagram or a Venn diagram) to organize these quadrilaterals: rhombus, rectangle, square, trapezoid. 90 CHAPTER 1 Introducing Geometry ● CHAPTER 1 REVIEW ● CHAPTER 1 REVIEW ● CHAPTER 1 REVIEW ● CHAPTER 1 REVIEW ● CHAPTER 1 9 in. 14 in. 48. The box at right is wrapped with two strips of ribbon, as shown. What is the minimum length of ribbon needed to decorate the box? 5 in. 49. At one point in a race, Rico was 15 ft behind Paul and 18 ft ahead of Joe. Joe was trailing George by 30 ft. Paul was ahead of George by how many ft? 50. A large aluminum ladder was resting vertically against the research shed at midnight when it began to slide down the side of the shed. A burglar was clinging to the ladder’s midpoint, holding a pencil flashlight that was visible in the dark. Witness Jill Seymour claimed to see the ladder slide. What did she see? That is, what was the path taken by the bulb of the flashlight? Draw a diagram showing the path. (Devise a physical test to check your visual thinking. You might try sliding a meterstick against a wall, or you might plot points on graph paper.) 51. Jiminey Cricket is caught in a windstorm. At 5:00 P.M. he is 500 cm away from his home. Each time he jumps toward home he leaps a distance of 50 cm, but before he regains strength to jump again he is blown back 40 cm. If it takes |
a full minute between jumps, how long will it take Jiminey to get home? 52. If the right triangle BAR were rotated 90° clockwise about point B, to what location would point A be relocated? 53. Sketch the three-dimensional figure formed by folding the net below into a solid. y R (2, 2) x A (–2, –1) B (2, –1) 54. Sketch the solid of revolution formed when you spin the two-dimensional figure about the line. 55. Sketch the section formed when the solid is sliced by the plane, as shown. CHAPTER 1 REVIEW 91 EW ● CHAPTER 1 REVIEW ● CHAPTER 1 REVIEW ● CHAPTER 1 REVIEW ● CHAPTER 1 REVIEW ● CHAPTE 56. Use an isometric dot grid to sketch the 57. Sketch the figure shown with the red edge figure shown below. vertical and facing the viewer. Assessing What You’ve Learned ORGANIZE YOUR NOTEBOOK Is this textbook filling up with folded-up papers stuffed between pages? If so, that’s a bad sign! But it’s not too late to get organized. Keeping a well-organized notebook is one of the best habits you can develop to improve and assess your learning. You should have sections for your classwork, definition list, and homework exercises. There should be room to make corrections and to summarize what you learned and write down questions you still have. Many books include a definition list (sometimes called a glossary) in the back. This book makes you responsible for your own glossary, so it’s essential that, in addition to taking good notes, you keep a complete definition list that you can refer to. You started a definition list in Lesson 1.1. Get help from classmates or your teacher on any definition you don’t understand. As you progress through the course, your notebook will become more and more important. A good way to review a chapter is to read through the chapter and your notes and write a one-page summary of the chapter. And if you create a one-page summary for each chapter, the summaries will be very helpful to you when it comes time for midterms and final exams. You’ll find no better learning and study aid than a summary page for each chapter, and your definition list, kept in an organized notebook. UPDATE YOUR PORTFOLIO If you did the project in this chapter, document your work and add it |
to your portfolio. Choose one homework assignment that demonstrates your best work in terms of completeness, correctness, and neatness. Add it (or a copy of it) to your portfolio. 92 CHAPTER 1 Introducing Geometry CHAPTER 2 Reasoning in Geometry That which an artist makes is a mirror image of what he sees around him. M. C. ESCHER Hand with Reflecting Sphere (Self-Portrait in Spherical Mirror), M. C. Escher ©2002 Cordon Art B. V.–Baarn–Holland. All rights reserved In this chapter you will ● perform geometry investigations and make many discoveries by observing common features or patterns ● use your discoveries to solve problems through a process called inductive reasoning ● use inductive reasoning to discover patterns ● learn to use deductive reasoning ● learn about vertical angles and linear pairs ● make conjectures L E S S O N 2.1 We have to reinvent the wheel every once in a while, not because we need a lot of wheels; but because we need a lot of inventors. BRUCE JOYCE Inductive Reasoning As a child you learned by experimenting with the natural world around you. You learned how to walk, to talk, and to ride your first bicycle, all by trial and error. From experience you learned to turn a water faucet on with a counterclockwise motion and to turn it off with a clockwise motion. You achieved most of your learning by a process called inductive reasoning. It is the process of observing data, recognizing patterns, and making generalizations about those patterns. Geometry is rooted in inductive reasoning. In ancient Egypt and Babylonia, geometry began when people developed procedures for measurement after much experience and observation. Assessors and surveyors used these procedures to calculate land areas and to reestablish the boundaries of agricultural fields after floods. Engineers used the procedures to build canals, reservoirs, and the Great Pyramids. Throughout this course you will use inductive reasoning. You will perform investigations, observe similarities and patterns, and make many discoveries that you can use to solve problems. Language The word “geometry” means “measure of the earth” and was originally inspired by the ancient Egyptians. The ancient Egyptians devised a complex system of land surveying in order to reestablish land boundaries that were erased each spring by the annual flooding of the Nile River. Inductive reasoning guides scientists, investors, and business managers. All of these professionals use past experience to assess what |
is likely to happen in the future. When you use inductive reasoning to make a generalization, the generalization is called a conjecture. Consider the following example from science. 94 CHAPTER 2 Reasoning in Geometry EXAMPLE A Solution A scientist dips a platinum wire into a solution containing salt (sodium chloride), passes the wire over a flame, and observes that it produces an orange-yellow flame. She does this with many other solutions that contain salt, finding that they all produce an orange-yellow flame. Make a conjecture based on her findings. The scientist tested many other solutions containing salt, and found no counterexamples. You should conjecture: “If a solution contains sodium chloride, then in a flame test it produces an orange-yellow flame.” Platinum wire flame test Like scientists, mathematicians often use inductive reasoning to make discoveries. For example, a mathematician might use inductive reasoning to find patterns in a number sequence. Once he knows the pattern, he can find the next term. EXAMPLE B Consider the sequence 2, 4, 7, 11,... Make a conjecture about the rule for generating the sequence. Then find the next three terms. Solution Look at the numbers you add to get each term. The 1st term in the sequence is 2. You add 2 to find the 2nd term. Then you add 3 to find the 3rd term, and so on. +2 +3 +4 2, 4, 7, 11 You can conjecture that if the pattern continues, you always add the next counting number to get the next term. The next three terms in the sequence will be 16, 22, and 29. +5 +6 +7 11, 16, 22, 29 In the following investigation you will use inductive reasoning to recognize a pattern in a series of drawings and use it to find a term much farther out in a sequence. LESSON 2.1 Inductive Reasoning 95 Investigation Shape Shifters Look at the sequence of shapes below. Pay close attention to the patterns that occur in every other shape. Step 1 Step 2 Step 3 Step 4 Step 5 What patterns do you notice in the 1st, 3rd, and 5th shapes? What patterns do you notice in the 2nd, 4th, and 6th shapes? Draw the next two shapes in the sequence. Use the patterns you discovered to draw the 25th shape. Describe the 30th shape in the sequence. You do not have to draw it! Sometimes a |
conjecture is difficult to find because the data collected are unorganized or the observer is mistaking coincidence with cause and effect. Good use of inductive reasoning depends on the quantity and quality of data. Sometimes not enough information or data have been collected to make a proper conjecture. For example, if you are asked to find the next term in the pattern 3, 5, 7, you might conjecture that the next term is 9—the next odd number. Someone else might notice that the pattern is the consecutive odd primes and say that the next term is 11. If the pattern was 3, 5, 7, 11, 13, what would you be more likely to conjecture? EXERCISES 1. On his way to the local Hunting and Gathering Convention, caveperson Stony Grok picks up a rock, drops it into a lake, and notices that it sinks. He picks up a second rock, drops it into the lake, and notices that it also sinks. He does this five more times. Each time, the rock sinks straight to the bottom of the lake. Stony conjectures: “Ura nok seblu,” which translates to?. What counterexample would Stony Grok need to find to disprove, or at least to refine, his conjecture? 96 CHAPTER 2 Reasoning in Geometry 2. Sean draws these geometric figures on paper. His sister Courtney measures each angle with a protractor. They add the measures of each pair of angles to form a conjecture. Write their conjecture. 90° 90° 150° 30° 45° 135° For Exercises 3–10, use inductive reasoning to find the next two terms in each sequence. 3. 1, 10, 100, 1000,?,? 5. 7, 3, 1, 5, 9, 13,?,? 7. 1, 1, 2, 3, 5, 8, 13,?,? 9. 32, 30, 26, 20, 12, 2. 1, 3, 6, 10, 15, 21,?,? 8. 1, 4, 9, 16, 25, 36,?,? 10. 1, 2, 4, 8, 16, 32,?,? For Exercises 11–16, use inductive reasoning to draw the next shape in each picture pattern. 11. 13. 15. 12. 14. 16. Use the rule provided to generate the first five terms of the sequence in Exercise 17 and the |
next five terms of the sequence in Exercise 18. 17. 3n 2 18. 1, 3, 6, 10,..., n(n 2 1),... 19. Now it’s your turn. Generate the first five terms of a sequence. Give the sequence to a member of your family or to a friend and ask him or her to find the next two terms in the sequence. Can he or she find your pattern? 20. Write the first five terms of two different sequences in which 12 is the 3rd term. 21. Think of a situation in which you have used inductive reasoning. Write a paragraph describing what happened and explaining why you think it was inductive reasoning. LESSON 2.1 Inductive Reasoning 97 22. Look at the pattern in these pairs of equations. Decide if the conjecture is true. If it is not true, find a counterexample. 122 144 132 169 1032 10609 1122 12544 and and and and 212 441 312 961 3012 90601 2112 44521 Conjecture: If two numbers have the same digits in reverse order, then the squares of those numbers will have identical digits but in reverse order. Review 23. Sketch the section formed when the cone is sliced by the plane, as shown. 24. 25. 26. Sketch the three-dimensional figure formed by folding the net below into a solid. 27. Sketch the figure shown 28. Sketch the solid of below but with the red edge vertical and facing you. revolution formed when the two-dimensional figure is rotated about the line. For Exercises 29–38, write the word that makes the statement true. 29. Points are? if they lie on the same line. 30. A triangle with two congruent sides is?. 31. The geometry tool used to measure the size of an angle in degrees is called a(n)?. 32. A(n)? of a circle connects its center to a point on the circle. 33. A segment connecting any two non-adjacent vertices in a polygon is called a(n)?. 34. A polygon with 12 sides is called a(n)?. 35. A trapezoid has exactly one pair of? sides. 98 CHAPTER 2 Reasoning in Geometry 36. A(n)? polygon is both equiangular and equilateral. 37. If angles are complementary, then their measures add to?. |
38. If two lines intersect to form a right angle, then they are?. For Exercises 39–42, sketch and label the figure. 39. Pentagon GIANT with diagonal AG parallel to side NT 40. A quadrilateral that has reflectional symmetry but not rotational symmetry 41. A prism with a hexagonal base 42. A counterexample to show that the following statement is false: The diagonals of a kite bisect the angles. IMPROVING YOUR REASONING SKILLS Puzzling Patterns These patterns are “different.” Your task is to find the next term. 1. 18, 46, 94, 63, 52, 61,? 2. O, T, T, F, F, S, S, E, N,? 3. 1, 4, 3, 16, 5, 36, 7,? 4. 4, 8, 61, 221, 244, 884,? 5. 6, 8, 5, 10, 3, 14, 1,? 6. B, 0, C, 2, D, 0, E, 3, F, 3, G,? 7. 2, 3, 6, 1, 8, 6, 8, 4, 8, 4, 8, 3, 2, 3, 2, 3,? 8 Where do the X, Y, and Z go? LESSON 2.1 Inductive Reasoning 99 Deductive Reasoning Have you ever noticed that the days are longer in the summer? Or that mosquitoes appear after a summer rain? Over the years you have made conjectures, using inductive reasoning, based on patterns you have observed. When you make a conjecture, the process of discovery may not always help explain why the conjecture works. You need another kind of reasoning to help answer this question. Deductive reasoning is the process of showing that certain statements follow logically from agreed-upon assumptions and proven facts. When you use deductive reasoning, you try to reason in an orderly way to convince yourself or someone else that your conclusion is valid. If your initial statements are true, and you give a logical argument, then you have shown that your conclusion is true. For example, in a trial, lawyers use deductive arguments to show how the evidence that they present proves their case. A lawyer might make a very good argument. But first, the court must believe the evidence and accept it as true. L E S S O N 2.2 That’s |
the way things come clear. All of a sudden. And then you realize how obvious they’ve been all along. MADELEINE L’ENGLE The success of an attorney’s case depends on the jury accepting the evidence as true and following the steps in her deductive reasoning. You use deductive reasoning in algebra. When you provide a reason for each step in the process of solving an equation, you are using deductive reasoning. Here is an example. EXAMPLE A Solve the equation for x. Give a reason for each step in the process. Solution 3(2x 1) 2(2x 1) 7 42 5x 3(2x 1) 2(2x 1) 7 42 5x 5(2x 1) 7 42 5x 5(2x 1) 35 5x 10x 5 35 5x 10x 30 5x 15x 30 x 2 The original equation. Combining like terms. Subtraction property of equality. Distributive property. Subtraction property of equality. Addition property of equality. Division property of equality. 100 CHAPTER 2 Reasoning in Geometry The next example shows how to use both kinds of reasoning: inductive reasoning to discover the property and deductive reasoning to explain why it works. EXAMPLE B In each diagram, AC bisects obtuse angle BAD. Classify BAD, DAC, and CAB as acute, right, or obtuse. Then complete the conjecture. C A B C B A D mBAD 120° A mBAD 158° D B D C mBAD 92° Conjecture: If an obtuse angle is bisected, then the two newly formed congruent angles are?. Justify your answers with a deductive argument. Solution In each diagram, BAD is obtuse because mBAD is greater than 90°. In each diagram, the angles formed by the bisector are acute because their measures— 60°, 79°, and 46°—are less than 90°. So one possible conjecture is Conjecture: If an obtuse angle is bisected, then the two newly formed congruent angles are acute. Why? According to our definition of an angle, every angle measure is less than 180°. So, using algebra, if m is the measure of an obtuse angle, then m 180°. When you bisect an angle, the two newly formed angles each measure half of the original |
angle, or m 1 m. If m 180°, then 1 1 (180), 2 2 2 so 1 m 90°. The two angles are each less 2 than 90°, so they are acute. m1_ 2 m m1_ 2 Science Here is an example of inductive reasoning, supported by deductive reasoning. El Niño is the warming of water in the tropical Pacific Ocean, which produces unusual weather conditions and storms worldwide. For centuries, farmers living in the Andes Mountains of South America have observed the stars in the Pleiades constellation to predict El Niño conditions. If the Pleiades look dim in June, they predict an El Niño year. What is the connection? Scientists have recently found that in an El Niño year, increased evaporation from the ocean produces high-altitude clouds that are invisible to the eye, but create a haze that makes stars more difficult to see. Therefore, the pattern that the Andean farmers knew about for centuries is now supported by a scientific explanation. To find out more about this story, go to www.keymath.com/DG. El Niño Conditions Increased convection Convective loop Equator Warm upper ocean layer Cold lower ocean layer 120°E 80°W LESSON 2.2 Deductive Reasoning 101 Inductive reasoning allows you to discover new ideas based on observed patterns. Deductive reasoning can help explain why your conjectures are true. Inductive and deductive reasoning work very well together. In this investigation you will use inductive reasoning to form a conjecture and deductive reasoning to explain why it’s true. Investigation Overlapping Segments In each segment, AB CD. Step 1 Step 2 Step 3 Step 4 25 cm 75 cm 25 cm 36 cm 80 cm 36 cm A B C D A B C D From the markings on each diagram, determine the lengths of AC and BD. What do you discover about these segments? Draw a new segment. Label it AD. Place your own points B and C on AD so that AB CD. A B C D Measure AC and BD. How do these lengths compare? Complete the conclusion of this conjecture: If AD has points A, B, C, and D in that order with AB CD, then?. Now you will use deductive reasoning and algebra to explain why the conjecture from Step 4 is true. Step 5 Use deductive reasoning to convince your group that AC will always equal BD. Take turns explaining to each other. Write your argument algebraically. In the investigation you |
used both inductive and deductive reasoning to convince yourself of the overlapping segments property. You will use a similar process in the next lesson to discover and prove the overlapping angles property in Exercise 17. Good use of deductive reasoning depends on the quality of the argument. Just like the saying, “A chain is only as strong as its weakest link,” a deductive argument is only as good (or as true) as the statements used in the argument. A conclusion in a deductive argument is true only if all the statements in the argument are true. Also, the statements in your argument must clearly follow from each other. Did you use clear arguments in explaining the investigation steps? Did you point out that BC is part of both AC and BD? Did you point out that if you add the same amount to things that are equal the resulting sum must be equal? 102 CHAPTER 2 Reasoning in Geometry EXERCISES 1. When you use? reasoning you are generalizing from careful observation that something is probably true. When you use? reasoning you are establishing that, if a set of properties is accepted as true, something else must be true. 2. A and B are complementary. mA 25°. What is mB? What type of reasoning do you use, inductive or deductive reasoning, when solving this problem? 3. If the pattern continues, what are the next two terms? What type of reasoning do you use, inductive or deductive reasoning, when solving this problem? 1 4 9 16 4. DGT is isosceles with TD DG. If the perimeter of DGT is 756 cm and GT 240 cm, then DG?. What type of reasoning do you use, inductive or deductive reasoning, when solving this problem? 5. Mini-Investigation The sum of the measures of the five marked angles in stars A through C is shown below each star. Use your protractor to carefully measure the five marked angles in star D. A B C D E 180° 180° 180°?? If this pattern continues, without measuring, what would be the sum of the measures of the marked angles in star E? What type of reasoning do you use, inductive or deductive reasoning, when solving this problem? 6. The definition of a parallelogram says, “If both pairs of opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram.” Quadrilateral LNDA has |
both pairs of opposite sides parallel. What conclusion can you make? What type of reasoning did you use? 7. Use the overlapping segments property to complete each statement. B C D A a. If AB 3, then CD?. b. If AC 10, then BD?. c. If BC 4 and CD 3, then AC?. 8. In Example B of this lesson you discovered through inductive reasoning that if an obtuse angle is bisected, then the two newly formed congruent angles are acute. You then used deductive reasoning to explain why they were acute. Go back to the example and look at the sizes of the acute angles formed. What is the smallest possible size for the two congruent acute angles formed by the bisector? Can you use deductive reasoning to explain why? LESSON 2.2 Deductive Reasoning 103 9. Study the pattern and make a conjecture by completing the fifth line. What would be the conjecture for the sixth line? The tenth line? 1 1 1 11 11 121 111 111 12,321 1,111 1,111 1,234,321 11,111 11,111? 10. Think of a situation you observed outside of school in which deductive reasoning was used correctly. Write a paragraph or two describing what happened and explaining why you think it was deductive reasoning. Review 11. Mark Twain once observed that the lower Mississippi River is very crooked and that over the years, as the bends and the turns straighten out, the river gets shorter and shorter. Using numerical data about the length of the lower part of the river, he noticed that in the year 1700 the river was more than 1200 miles long, yet by the year 1875 it was only 973 miles long. Twain concluded that any person “can see that 742 years from now the lower Mississippi will be only a mile and three-quarters long.” What is wrong with this inductive reasoning? For Exercises 12–14, use inductive reasoning to find the next two terms of the sequence. 12. 180, 360, 540, 720,?,? 13. 0, 10, 21, 33, 46, 60,?,?, 10, 3, 9, 2 14. 1, 11,?,? 4 3 2 For Exercises 15–18, draw the next shape in each picture pattern. 15. 17. 16. 18. Aerial photo of the Mississippi River 19. Think of a situation you have observed in which induct |
ive reasoning was used incorrectly. Write a paragraph or two describing what happened and explaining why you think it was an incorrect use of inductive reasoning. 104 CHAPTER 2 Reasoning in Geometry Match each term in Exercises 20–29 with one of the figures A–O. 20. Kite 22. Trapezoid 24. Pair of complementary angles 26. Pair of vertical angles 28. Acute angle A. F. K. 48° 132°?? B. G. L. 21. Consecutive angles in a polygon 23. Diagonal in a polygon 25. Radius 27. Chord 29. Angle bisector in a triangle C.? D.? O 89° H. M. I. N.?? 124° E. 52° 38° J.? O. For Exercises 30–33, sketch and carefully label the figure. 30. Pentagon WILDE with ILD LDE and LD DE 31. Isosceles obtuse triangle OBG with mBGO 140° 32. Circle O with a chord CD perpendicular to radius OT 33. Circle K with angle DIN where D, I, and N are points on circle K IMPROVING YOUR VISUAL THINKING SKILLS Rotating Gears In what direction will gear E rotate if gear A rotates in a counterclockwise direction? B D E F A C LESSON 2.2 Deductive Reasoning 105 L E S S O N 2.3 If you do something once, people call it an accident. If you do it twice, they call it a coincidence. But do it a third time and you’ve just proven a natural law. GRACE MURRAY HOPPER Finding the nth Term What would you do to get the next term in the sequence 20, 27, 34, 41, 48, 55,...? A good strategy would be to find a pattern, using inductive reasoning. Then you would look at the differences between consecutive terms and predict what comes next. In this case there is a constant difference of 7. That is, you add 7 each time. The next term is 55 7, or 62. What if you needed to know the value of the 200th term of the sequence? You certainly don’t want to generate the next 193 terms just to get one answer. If you knew a rule for calculating any term in a sequence, without having to know the previous term, you could apply it to directly |
calculate the 200th term. The rule that gives the nth term for a sequence is called the function rule. Let’s see how the constant difference can help you find the function rule for some sequences. DRABBLE reprinted by permission of United Feature Syndicate, Inc. Investigation Finding the Rule Step 1 Copy and complete each table. Find the differences between consecutive values. a. b. c. d. e 4n 3 n 2n 5 n 3n 5n 106 CHAPTER 2 Reasoning in Geometry Step 2 Did you spot the pattern? If a sequence has a constant difference of 4, then the number in front of the n (the coefficient of n) is?. In general, if the difference between the values of consecutive terms of a sequence is always the same, say m (a constant), then the coefficient of n in the formula is?. Let’s return to the sequence at the beginning of the lesson. Term Value 1 20 2 27 3 34 4 41 5 48 6 55 7 … n 62 … 7 7 The constant difference is 7, so you know part of the rule is 7n. How do you find the rest of the rule? Step 3 The first term (n 1) of the sequence is 20, but if you apply the part of the rule you have so far using n 1, you get 7n 7(1) 7, not 20. So how should you fix the rule? How can you get from 7 to 20? What is the rule for this sequence? Step 4 Check your rule by trying the rule with other terms in the sequence. Let’s look at an example of how to find a function rule, for the nth term in a number pattern. EXAMPLE A Find the rule for the sequence 7, 2, 3, 8, 13, 18,... Solution Placing the terms and values in a table we get Term Value 13 18 … 6 n The difference between the terms is always 5. So the rule is 5n “something” Let’s use c to stand for the unknown “something.” So the rule is 5n c To find c, replace the n in the rule with a term number. Try n 1 and set the expression equal to 7. 5(1) c 7 c 12 The rule is 5n 12. You can find the value of any term in the sequence by substituting the term number for n in the function rule. Let’s look |
at an example of how to find the 200th term in a geometric pattern. LESSON 2.3 Finding the nth Term 107 EXAMPLE B If you place 200 points on a line, into how many non-overlapping rays and segments does it divide the line? Solution Wait! don’t start placing 200 points on a line. You need to find a rule that relates the number of points placed on a line to the number of parts created by those points. Then you can use your rule to answer the problem. Start by creating a table. Points dividing the line 1 2 3 4 5 6 … n … 200 Non-overlapping rays Non-overlapping segments Total … … … … … … Sketch one point dividing a line. One point gives you just two rays. Enter that into the table. Next, sketch two points dividing a line. This gives one segment and the two end rays. Enter the value into your table. Next, sketch three points dividing a line, then four, then five, and so on. The table completed for one to three points is Points dividing the line Non-overlapping rays Non-overlapping segments Total … 200 … … … … … … Once you have found values for 1, 2, 3, 4, 5, and 6 points on a line you next try to find the rule for each sequence. There are always two non-overlapping rays so for 200 points there will be two rays. The rule, or nth term, for the number of non-overlapping segments is n 1. For 200 points there will be 199 segments. The rule, or nth term, for the total number of distinct rays and segments of the line is n 1. For 200 points there will be 201 distinct parts of the line. This process of looking at patterns and generalizing a rule, or nth term, is inductive reasoning. To understand why the rule is what it is, you can turn to deductive reasoning. Notice that adding another point on a line divides a segment into two segments. So each new point adds one more segment to the pattern. Rules that generate a sequence with a constant difference are linear functions. To see why they’re called linear, you can graph the term number and the value for the sequence as ordered pairs of the form (term number, value) on the coordinate plane. At left is the graph of the sequence from Example A. Term number n Value f (n 13 18 … |
5n 12 … n 5 f(n) 10 Term number 5 –10 –15 –20 108 CHAPTER 2 Reasoning in Geometry EXERCISES For Exercises 1–3, find the function rule f(n) for each sequence. Then find the 20th term in the sequence. 1. 2. 3. n f (n) n f (n) 1 3 1 1 2 9 3 15 4 21 5 27 6 … 33 … 3 2 … 2 5 8 11 14 … 5 4 6 1 n f (n) 4 2 4 3 12 4 20 5 28 6 … 36 … n n n … 20 … … 20 … … 20 … For Exercises 4–6, find the rule for the nth figure. Then find the number of colored tiles or matchsticks in the 200th figure. 4. 5. 6. Figure number Number of tiles … 200 … Figure number 1 Number of tiles 2 5 3 4 5 6 … … n … 200 … Figure number Number of matchsticks Number of matchsticks in perimeter of figure … 200 … … LESSON 2.3 Finding the nth Term 109 7. How many triangles are formed when you draw all the possible diagonals from just one vertex of a 35-gon? Number of sides 3 4 5 6 Number of triangles formed … … n … 35 … 8. Graph the values in your tables from Exercises 4–6. Which set of points lies on a steeper line? What number in the rule gives a measure of steepness? 9. Find the rule for the set of points in the graph shown at right. Place the x-coordinate of each ordered pair in the top row of your table and the corresponding y-coordinate in the second row. What is the value of y in terms of x? Review For Exercises 10–13, sketch and carefully label the figure. 10. Equilateral triangle EQL with QT where T lies on EL and QT EL 11. Isosceles obtuse triangle OLY with OL YL and angle bisector LM y (4, 9) (2, 6) (0, 3) (–2, 0) (–4, –3) x 12. A cube with a plane passing through it; the cross section is rectangle RECT 13. A net for a rectangular solid with the dimensions 1 by 2 by 3 cm 14. Márisol’s younger brother José was drawing triangles when he noticed |
that every triangle he drew turned out to have two sides congruent. José conjectures: “Look, Márisol, all triangles are isosceles.” How should Márisol respond? 15. A midpoint divides a segment into two congruent segments. Point M divides segment AY into two congruent segments AM and MY. What conclusion can you make? What type of reasoning did you use? 16. Tanya’s favorite lunch is peanut butter and jelly on wheat bread with a glass of milk. Lately, she has been getting an allergic reaction after eating this lunch. She is wondering if she might be developing an allergy to peanut butter, wheat, or milk. What experiment could she do to find out which food it is? What type of reasoning would she be using? 110 CHAPTER 2 Reasoning in Geometry 17. Mini-Investigation Do the geometry investigation and make a conjecture. A Given APB with points C and D in its interior and mAPC mDPB, If mAPD 48°, then mCPB? If mCPB 17°, then mAPD? If mAPD 62°, then mCPB? Conjecture: If points C and D lie in the interior of APB, and mAPC mDPB then mAPD? (Overlapping angles property) D C B P With Fathom Dynamic Statistics™ software, you can plot your data points and find the linear equation that best fits your data. BEST-FIT LINES The following table and graph show the mileage and lowest priced round-trip airfare between New York City and each destination city. Is there a relationship between the money you spend and how far you can travel? Lowest Round-trip Airfares from New York City on February 25, 2002 Destination City Distance (miles) Price ($) Boston Chicago Atlanta Miami Denver Phoenix Los Angeles Source: http://www.Expedia.com 215 784 865 1286 1791 2431 2763 $118 $178 $158 $170 $238 $338 $298 Even though the data are not linear, you can find a linear equation that approximately fits the data. The graph of this equation is called the line of best fit. How would you use the line of best fit to predict the cost of a round-trip ticket to Seattle (2814 miles)? How would you use it to determine how far you could travel (in miles) with $250? How accurate do |
you think the answer would be? Choose a topic and a relationship to explore. You can use data from the census (such as age and income), or data you collect yourself (such as number of ice cubes in a. glass and melting time). For more sources and ideas, go to www.keymath.com/DG Collect pairs of data points. Use Fathom to graph your points and to find the line of best fit. Write a summary of your results. LESSON 2.3 Finding the nth Term 111 L E S S O N 2.4 It’s amazing what one can do when one doesn’t know what one can’t do. GARFIELD THE CAT Mathematical Modeling Physical models have many of the same features as the original object or activity they represent, but are often more convenient to study. For example, building a new airplane and testing it is difficult and expensive. But you can analyze a new airplane design by building a model and testing it in a wind tunnel. In Chapter 1 you learned that geometry ideas such as points, lines, planes, triangles, polygons, and diagonals are mathematical models of physical objects. When you draw graphs or pictures of situations, or when you write equations that describe a problem, you are creating mathematical models. A physical model of a complicated telecommunications network, for example, might not be practical, but you can draw a mathematical model of the network using points and lines. This computer model tests the effectiveness of the car’s design for minimizing wind resistance. This computer-generated model uses points and line segments to show the volume of data traveling to different locations on the National Science Foundation Network. In this investigation, you will attempt to solve a problem first by acting it out, then by creating a mathematical model. Investigation Party Handshakes Each of the 30 people at a party shook hands with everyone else. How many handshakes were there altogether? Step 1 Act out this problem with members of your group. Collect data for “parties” of one, two, three, and four people and record your results in a table. People Handshakes 1 0 2 1 3 4 … 30 … 112 CHAPTER 2 Reasoning in Geometry Step 2 Look for a pattern. Can you generalize from your pattern to find the 30th term? Acting out a problem is a powerful problem-solving strategy that can give you important insight into a solution. Were you able to make a generalization from just four |
terms? If so, how confident are you of your generalization? To collect more data, you can ask more classmates to join your group. You can see, however, that acting out a problem sometimes has its practical limitations. That’s where you can use mathematical models. Step 3 Model the problem by using points to represent people and line segments connecting the points to represent handshakes. Record your results in a table like this one: Number of points (people) Number of segments (handshakes … 30 … … Notice that the pattern does not have a constant difference. That is, the rule is not a linear function. So we need to look for a different kind of rule. 2 2 2 3 3 3 3 3 points 2 segments per vertex 4 points 3 segments per vertex 4 4 4 4 4 5 points segments per? vertex 5 5 5 5 5 5 6 points segments per? vertex LESSON 2.4 Mathematical Modeling 113 Step 4 Refer to the table you made for Step 3. The pattern of differences is increasing by one: 1, 2, 3, 4, 5, 6, 7. Read the dialogue between Erin and Stephanie as they attempt to combine inductive and deductive reasoning to find the rule. In the diagram with 3 vertices there are 2 segments from each vertex. If there are 2 segments from each of the 3 vertices, why isn’t the rule 2 3, or 6 segments? Because you are counting each segment twice, the answer is really 3 2 2, or 3 segments. Right, but each segment got counted twice. So divide by 2. So, in the diagram with 4 vertices there are 3 segments from each vertex… …so there 3, or are 4 2 6 segments. 3 3 2 3 2 ____ 2 Let’s continue with Stephanie and Erin’s line of reasoning. Step 5 In the diagram with 5 vertices how many segments are there from each vertex? So the total number of segments?. written in factored form is 5 2 4 4 3 4 3 ____ 2 Step 6 Complete the table below by expressing the total number of segments in factored form. Number of points (people) 1 2 3 4 5 6 Number of segments (1) (?) (?) (6) (3) (5) (2) (4) (1) (3) (0) (2) (handshakes (?) (?) 2 Step 7 The larger of the two factors in the numerator |
represents the number of points. What does the smaller of the two numbers in the numerator represent? Why do we divide by 2? Step 8 Write a function rule. How many handshakes were there at the party? 114 CHAPTER 2 Reasoning in Geometry The numbers in the pattern in the previous investigation are called the triangular numbers because you can arrange them into a triangular pattern of dots. 1 3 6 10 Fifteen pool balls can be arranged in a triangle, so 15 is a triangular number. The triangular numbers appear in many geometric situations. You will see some of them in the exercises. Here is a visual approach to arrive at the rule for this special pattern of numbers. If we arrange the triangular numbers in stacks, 1 3 6 10 you can see that each is half of a rectangular number. To get the total number of dots in a rectangular array, you multiply the number of rows by the number of dots in each row. In the case of this rectangular array, the nth rectangle has n(n 1) dots. So, the triangular array has n(n 2 1) dots. EXERCISES For Exercises 1–6, draw the next figure. Complete a table and find the function rule. Then find the 35th term. 1. Lines passing through the same point are concurrent. Into how many regions do 35 concurrent lines divide the plane? Lines Regions 1 2 2 3 4 5 … … n … 35 … 2. Into how many regions do 35 parallel lines in a plane divide that plane? LESSON 2.4 Mathematical Modeling 115 3. How many diagonals can you draw from one vertex in a polygon with 35 sides? 4. What’s the total number of diagonals in a 35-sided polygon? 5. If you place 35 points on a piece of paper so that no three points are in a line, how many line segments are necessary to connect each point to all the others? 6. If you draw 35 lines on a piece of paper so that no two lines are parallel to each other and no three lines are concurrent, how many times will they intersect? 7. Look at the formulas you found in Exercises 4–6. Describe how the formulas are related. Then explain how the three problems are related geometrically. For Exercises 8–10, draw a diagram, find the appropriate geometric model, and solve. 8. If 40 houses in a community all need direct lines to one another in |
order to have telephone service, how many lines are necessary? Is that practical? Sketch and describe two models: first, model the situation in which direct lines connect every house to every other house and, second, model a more practical alternative. 9. If each team in a ten-team league plays each of the other teams four times in a season, how many league games are played during one season? What geometric figures can you use to model teams and games played? 10. Each person at a party shook hands with everyone else exactly once. There were 66 handshakes. How many people were at the party? Review For Exercises 11–19, identify the statement as true or false. For each false statement, explain why it is false or sketch a counterexample. 11. The largest chord of a circle is a diameter of the circle. 116 CHAPTER 2 Reasoning in Geometry 12. The vertex of TOP is point O. 13. An isosceles right triangle is a triangle with an angle measuring 90° and no two sides congruent. 14. If AB intersects CD in point E, then AED and BED form a linear pair of angles. 15. If two lines lie in the same plane and are perpendicular to the same line, they are perpendicular. 16. The opposite sides of a kite are never parallel. 17. A rectangle is a parallelogram with all sides congruent. 18. A line segment that connects any two vertices in a polygon is called a diagonal. 19. To show that two lines are parallel, you mark them with the same number of arrowheads. 20. Hydrocarbons are molecules that consist of carbon (C) and hydrogen (H). Hydrocarbons in which all the bonds between the carbon atoms are single bonds are called alkanes. The first four alkanes are modeled below. Sketch the alkane with eight carbons in the chain. What is the general rule for alkanes CnH? hydrogen atoms (H) are in the alkane?? In other words, if there are n carbon atoms (C), how many Methane CH4 H H Ethane C2H6 H H H H H H H Propane C3H8 Butane C4H10 Science Organic chemistry is the study of carbon compounds and their reactions. Drugs, vitamins, synthetic fibers, and food all contain organic molecules. Organic chemists continue to improve our quality of life by the advances they make in medicine, nutrition |
, and manufacturing. To learn about new advances www.keymath.com/DG in organic chemistry, go to. IMPROVING YOUR VISUAL THINKING SKILLS Pentominoes II In Pentominoes I, you found the 12 pentominoes. Which of the 12 pentominoes can you cut along the edges and fold into a box without a lid? Here is an example. LESSON 2.4 Mathematical Modeling 117 L E S S O N 1.0 The Seven Bridges of Königsberg The River Pregel runs through the university town of Königsberg (now Kaliningrad in Russia). In the middle of the river are two islands connected to each other and Leonhard Euler to the rest of the city by seven bridges. Many years ago, a tradition developed among the townspeople of Königsberg. They challenged one another to make a round trip over all seven bridges, walking over each bridge once and only once before returning to the starting point. The seven bridges of Königsberg For a long time no one was able to do it, and yet no one was able to show that it couldn’t be done. In 1735, they finally wrote to Leonhard Euler (1707–1783), a Swiss mathematician, asking for his help on the problem. Euler (pronounced “oyler”) reduced the problem to a network of paths connecting the two sides of the rivers C and B, and the two islands A and D, as shown in the network at right. Then Euler demonstrated that the task is impossible. In this activity you will work with a variety of networks to see if you can come up with a rule to find out whether a network can or cannot be “traveled.” Activity Traveling Networks A collection of points connected by paths is called a network. When we say a network can be traveled, we mean that the network can be drawn with a pencil without lifting the pencil off the paper and without retracing any paths. (Points can be passed over more than once.) 118 CHAPTER 2 Reasoning in Geometry Step 1 Try these networks and see which ones can be traveled and which are impossible to travel. A. F. K. B. G. C. D. E. H. I. J. L. M. N. O. P. Which networks were impossible to travel? Are they impossible or just difficult? How |
can you be sure? As you do the next few steps, see if you can find the reason why some networks are impossible to travel. Step 2 Step 3 Draw the River Pregel and the two islands shown on the first page of this exploration. Draw an eighth bridge so that you can travel over all the bridges exactly once if you start at point C and end at point B. Draw the River Pregel and the two islands. Can you draw an eighth bridge so that you can travel over all the bridges exactly once, starting and finishing at the same point? How many solutions can you find? Step 4 Euler realized that it is the points of intersection that determine whether a network can be traveled. Each point of intersection is either “odd” or “even.” Odd points Even points Did you find any networks that have only one odd point? Can you draw one? Try it. How about three odd points? Or five odd points? Can you create a network that has an odd number of odd points? Explain why or why not. Step 5 How does the number of even points and odd points affect whether a network can be traveled? Conjecture A network can be traveled if?. EXPLORATION The Seven Bridges of Königsberg 119 L E S S O N 2.5 Discovery consists of looking at the same thing as everyone else and thinking something different. ALBERT SZENT-GYÖRGYI Angle Relationships Now that you’ve had experience with inductive reasoning, let’s use it to start discovering geometric relationships. This investigation is the first of many investigations you will do using your geometry tools. Create an investigation section in your notebook. Include a title and illustration for each investigation and write a statement summarizing the results of each one. Investigation 1 The Linear Pair Conjecture You will need ● a protractor S P R Q Step 1 On a sheet of paper, draw PQ and place a point R between P and Q. Choose another point S not on PQ and draw RS. You have just created a linear pair of angles. Place the “zero edge” of your protractor along PQ. What do you notice about the sum of the measures of the linear pair of angles? Step 2 Compare your results with those of your group. Does everyone make the same observation? Complete the statement. Linear Pair Conjecture If two angles form a linear pair, then?. C-1 The important conjectures have |
been given a name and a number. Start a list of them in your notebook. The Linear Pair Conjecture (C-1) and the Vertical Angles Conjecture (C-2) should be the first entries on your list. Make a sketch for each conjecture. 120 CHAPTER 2 Reasoning in Geometry In the previous investigation you discovered the relationship between a linear pair of angles, such as 1 and 2 in the diagram at right. You will discover the relationship between vertical angles, such as 1 and 3, in the next investigation. 1 2 4 3 Investigation 2 Vertical Angles Conjecture You will need ● a protractor ● patty paper 4 1 2 3 Step 1 Step 2 Draw two intersecting lines onto patty paper or tracing paper. Label the angles as shown. Which angles are vertical angles? Fold the paper so that the vertical angles lie over each other. What do you notice about their measures? Step 3 Repeat this investigation with another pair of intersecting lines. Step 4 Compare your results with the results of others. Complete the statement. Vertical Angles Conjecture If two angles are vertical angles, then?. C-2 You used inductive reasoning to discover both the Linear Pair Conjecture and the Vertical Angles Conjecture. Are they related in any way? That is, if we accept the Linear Pair Conjecture as true, can we use deductive reasoning to show that the Vertical Angles Conjecture must be true? EXAMPLE The Linear Pair Conjecture states that every linear pair adds up to 180°. Using this conjecture and the diagram, write a logical argument explaining why 1 must be congruent to 3. 2 4 3 1 LESSON 2.5 Angle Relationships 121 Solution You can see that the measures of 1 and 2 add up to 180°, and that the measures of 3 and 2 also add up to 180°. Using algebra, we can write a logical argument to show that 1 and 3 must be congruent. 180° 180° 2 4 3 1 According to the Linear Pair Conjecture, m1 m2 180° and m2 m3 180°. By substituting m2 m3 for 180° in the first statement, you get m1 m2 m2 m3. By the subtraction property of equality, you can subtract m2 from both sides of the equation to get m1 m3. Therefore, vertical angles 1 and 3 have equal measures and are congruent. Here are |
the algebraic steps: m2 m3 180° m1 m2 180° m1 m2 m2 m3 thus m1 m3 therefore 1 3 This type of logical explanation, written as a paragraph, is called a paragraph proof. Now consider another idea. You discovered the Vertical Angles Conjecture: If two angles are vertical angles, then they are congruent. Does that also mean that all congruent angles are vertical angles? The converse of an “if-then” statement switches the “if ” and “then” parts. The converse of the Vertical Angles Conjecture may be stated: If two angles are congruent, then they are vertical angles. Is this converse statement true? Remember that if you can find even one counterexample, like the diagram below, then the statement is false. Therefore, the converse of the Vertical Angles Conjecture is false. EXERCISES Without using a protractor, but with the aid of your two new conjectures, find the measure of each lettered angle in Exercises 1–5. Copy the diagrams so that you can write on them. List your answers in alphabetical order. 1. 2. 3. You will need Geometry software for Exercise 12 a b c 60° a b c 40° 51° b a c 52° d 122 CHAPTER 2 Reasoning in Geometry 4. 60° b c a 5. a c b 163° 70° d e 65° d e f h i g 55° 6. Points A, B, and C at right are collinear. What’s wrong with this picture? 129° 41° B C A 7. Yoshi is building a cold frame for his plants. He wants to cut two wood strips so that they’ll fit together to make a right-angled corner. At what angle should he cut ends of the strips??? 8. A tree on a 30° slope grows straight up. What are the measures of the greatest and smallest angles the tree makes with the hill? Explain. 30°?? 9. You discovered that if a pair of angles is a linear pair then the angles are supplementary. Does that mean that all supplementary angles form a linear pair of angles? Is the converse true? If not, sketch a counterexample. 10. If two congruent angles are supplementary, what must be true of the two angles? Make a |
sketch, then complete the following conjecture: If two angles are both congruent and supplementary, then?. 11. Using algebra, write a paragraph proof that explains why the conjecture from Exercise 10 is true. 12. Technology Use geometry software to construct two intersecting lines. Measure a pair of vertical angles. Use calculate to find the ratio of their measures. What is the ratio? Drag one of the lines. Does the ratio ever change? Does this demonstration convince you that the Vertical Angles Conjecture is true? Does it explain why it is true? Review For Exercises 13–17, sketch, label, and mark the figure. 13. Scalene obtuse triangle PAT with PA 3 cm, AT 5 cm, and A an obtuse angle 14. A quadrilateral that has rotational symmetry but not reflectional symmetry 15. A circle with center at O and radii OA and OT creating a minor arc AT LESSON 2.5 Angle Relationships 123 16. A pyramid with an octagonal base 17. A 3-by-4-by-6-inch rectangular solid rests on its smallest face. Draw lines on the three visible faces, showing how you can divide it into 72 identical smaller cubes. 18. Miriam the Magnificent placed four cards face up (the first four cards shown below). Blindfolded, she asked someone from her audience to come up to the stage and turn one card 180°. Before turn After turn Miriam removed her blindfold and claimed she was able to determine which card was turned 180°. What is her trick? Can you figure out which card was turned? Explain. 19. If a pizza is cut into 16 congruent pieces, how many degrees are in each angle at the center of the pizza? 20. Paulus Gerdes, a mathematician from Mozambique, uses traditional lusona patterns from Angola to practice inductive thinking. Shown below are three sona designs. Sketch the fourth sona design, assuming the pattern continues. 21. Hydrocarbon molecules in which all the bonds between the carbon atoms are single bonds except one double bond are called alkenes. The first three alkenes are modeled below Ethene C2H4 H H H H H H H Propene C3H6 Butene C4H8 Sketch the alkene with eight carbons in the chain. What is the general rule for alkenes CnH? hydrogen atoms (H) are in the alkene |
?? In other words, if there are n carbon atoms (C), how many 124 CHAPTER 2 Reasoning in Geometry 22. If the pattern of rectangles continues, what is the rule for the perimeter of the nth rectangle, and what is the perimeter of the 200th rectangle? Perimeter in a rectangular pattern Rectangle Perimeter of rectangle 1 10 2 14 3 18 4 5 6 … … n … 200 … 23. The twelfth grade class of 80 students is assembled in a large circle on the football field at halftime. Each student is connected by a string to each of the other class members. How many pieces of string are necessary to connect each student to all the others? 24. If you draw 80 lines on a piece of paper so that no 2 lines are parallel to each other and no 3 lines pass through the same point, how many intersections will there be? 25. If there are 20 couples at a party, how many different handshakes can there be between pairs of people? Assume that the two people in each couple do not shake hands with each other. 26. If a polygon has 24 sides, how many diagonals are there from each vertex? How many diagonals are there in all? 27. If a polygon has a total of 560 diagonals, how many vertices does it have? IMPROVING YOUR ALGEBRA SKILLS Number Line Diagrams 1. The two segments at right have the same length. x 3 x 3 x 3 20 Translate the number line diagram into an equation, then solve for the variable x. 2x 23 2x 23 30 2. Translate this equation into a number line diagram. 2(x 3) 14 3(x 4) 11 LESSON 2.5 Angle Relationships 125 L E S S O N 2.6 The greatest mistake you can make in life is to be continually fearing that you will make one. ELLEN HUBBARD Special Angles on Parallel Lines A line intersecting two or more other lines in the plane is called a transversal. A transversal creates different types of angle pairs. Three types are listed below. One pair of corresponding angles is 1 and 5. Can you find three more pairs of corresponding angles? Transversal 1 2 3 4 5 6 7 8 One pair of alternate interior angles is 3 and 6. Do you see another pair of alternate interior angles? One pair of alternate exterior angles is 2 and 7. Do you |
see the other pair of alternate exterior angles? When parallel lines are cut by a transversal, there is a special relationship among the angles. Let’s investigate. You will need ● lined paper or a straightedge ● patty paper ● a protractor Step 1 Investigation 1 Which Angles Are Congruent? Using the lines on your paper as a guide, draw a pair of parallel lines. Or use both edges of your ruler or straightedge to create parallel lines. Label them k and. Now draw a transversal that intersects the parallel lines. Label the transversal m, and label the angles with numbers, as shown at right. m 1 2 43 5 6 87 k Place a piece of patty paper over the set of angles 1, 2, 3, and 4. Copy the two intersecting lines m and and the four angles onto the patty paper. Slide the patty paper down to the intersection of lines m and k, and compare angles 1 through 4 with each of the corresponding angles 5 through 8. What is the relationship between corresponding angles? Alternate interior angles? Alternate exterior angles? 126 CHAPTER 2 Reasoning in Geometry Compare your results with the results of others in your group and complete the three conjectures below. Corresponding Angles Conjecture, or CA Conjecture If two parallel lines are cut by a transversal, then corresponding angles are?. 2 6 Alternate Interior Angles Conjecture, or AIA Conjecture If two parallel lines are cut by a transversal, then alternate interior angles are?. 3 6 Alternate Exterior Angles Conjecture, or AEA Conjecture If two parallel lines are cut by a transversal, then alternate exterior angles are?. 1 8 C-3a C-3b C-3c The three conjectures you wrote can all be combined to create a Parallel Lines Conjecture, which is really three conjectures in one. Parallel Lines Conjecture If two parallel lines are cut by a transversal, then corresponding angles are?, alternate interior angles are?, and alternate exterior angles are?. C-3 LESSON 2.6 Special Angles on Parallel Lines 127 Step 2 What happens if the lines you start with are not parallel? Check whether your conjectures will work with nonparallel lines. Corresponding angles Corresponding angles What about the converse of each of your conjectures? Suppose you know that a pair of corresponding angles, or alternate interior angles |
, is congruent. Will the lines be parallel? Is it possible for the angles to be congruent but for the lines not to be parallel? Investigation 2 Is the Converse True? You will need ● lined paper or a straightedge ● patty paper ● a protractor Step 1 Draw two intersecting lines on your paper. Copy these lines onto a piece of patty paper. Because you copied the angles, the two sets of angles are congruent. Slide the top copy so that the transversal stays lined up. Trace the lines and the angles from the bottom original onto the patty paper again. When you do this, you are constructing sets of congruent corresponding angles. Mark the congruent angles. Step 2 Are the two lines parallel? You can test to see if the distance between the two lines remains the same, which guarantees that they will never meet. Repeat Step 1, but this time rotate your patty paper 180° so that the transversal lines up again. What kinds of congruent angles have you created? Trace the lines and angles and mark the congruent angles. Are the lines parallel? Check them. 3 4 12 1 2 4 3 128 CHAPTER 2 Reasoning in Geometry Step 3 Compare your results with those of your group. If your results do not agree, discuss them until you have convinced each other. Complete the conjecture below and add it to your conjecture list. Converse of the Parallel Lines Conjecture C-4 If two lines are cut by a transversal to form pairs of congruent corresponding angles, congruent alternate interior angles, or congruent alternate exterior angles, then the lines are?. You used inductive reasoning to discover all three parts of the Parallel Lines Conjecture. However, if you accept any one of them as true, you can use deductive reasoning to show that the others are true. EXAMPLE Suppose we assume that the Vertical Angles Conjecture is true. Write a paragraph proof showing that if corresponding angles are congruent, then the Alternate Interior Angles Conjecture is true. Paragraph Proof Lines and m are parallel and intersected by transversal k. Pick any two alternate interior angles, such as 2 and 3. According to the Corresponding Angles Conjecture, 2 1. And, according to the Vertical Angles Conjecture, 1 3. Substitute 3 for 1 in the first statement to get 2 3. But 2 and 3 are |
alternate interior angles. Therefore, if the corresponding angles are congruent, then the alternate interior angles are congruent. Solution It helps to visualize each statement and to mark all congruences you know on your paper. EXERCISES k m 3 1 2 Here are the algebraic steps: 2 1 3 1 So 2 3 Use your new conjectures in Exercises 1–6. A small letter in an angle represents the angle measure. 1. r s w? 2. p q x? 3. Is line k parallel to line? 63° w r s p q x 68° 122° k LESSON 2.6 Special Angles on Parallel Lines 129 4. Quadrilateral TUNA is a parallelogram. y? A 5. Is quadrilateral FISH a parallelogram? 6. m n z? N y H 65° S 57° T U 115° F 65° I 67° 7. Trace the diagram below. Calculate each lettered angle measure. z m n c d 64° ba 75° m q p n 79° s k i j gh 108° f e 169° t 61° 8. You’ve seen before how parallel lines appear to meet in the distance. Let’s look at the converse of this effect: The top and the bottom of the Vietnam Veterans Memorial Wall appear to be parallel because they appear to meet so far in the distance. Consider the diagram of the corner of the memorial, shown below. You know that line 2 eventually meet. Is the blue shaded portion of the wall a rectangle? Write a paragraph proof explaining why it is or is not a rectangle. 1 and line Top 1 2 Bottom Top a b Corner Bottom Sculptor Maya Lin designed the Vietnam Veterans Memorial Wall in Washington, D.C. Engraved in the granite wall are the names of United States armed forces service members who died in the Vietnam War or remain missing in action. To learn more about the Memorial Wall and Lin’s other projects, visit www.keymath.com/DG. 130 CHAPTER 2 Reasoning in Geometry 9. What’s wrong with this picture? 10. What’s wrong with this picture? 56° 114° 45° 55° 55° 11. A periscope permits a sailor on a submarine to see above the surface of the ocean. This periscope is designed so that the line of sight a is parallel to the light ray b. The |
middle tube is perpendicular to the top and bottom tubes. What are the measures of the incoming and outgoing angles formed by the light rays and the mirrors in this periscope? Are the surfaces of the mirrors parallel? How do you know? Review 12. What type (or types) of triangle has one or more lines of symmetry? 13. What type (or types) of quadrilateral has only rotational symmetry? 14. If D is the midpoint of AC and C is the midpoint of BD, what is the length of AB if BD 12 cm? 15. If AI is the angle bisector of KAN and AR is the angle bisector of KAI, what is mRAN if mRAK 13°? For Exercises 16–18, draw each polygon on graph paper. Relocate the vertices according to the rule. Connect the new points to form a new polygon. Describe what happened to the figure. Is the new polygon congruent to the original? b a A K R I N 16. Rule: Subtract 1 from each x-coordinate. y 5 x 5 17. Rule: Reverse the sign of each x- and y-coordinate. y 4 x 4 18. Rule: Switch the x- and y-coordinates. Pentagon LEMON with vertices: L(4, 2) E(4, 3) M(0, 5) O(3, 1) N(1, 4) 19. If everyone in the town of Skunk’s Crossing (population 84) has a telephone, how many different lines are needed to connect all the phones to each other? 20. How many squares of all sizes are in a 4-by-4 grid of squares? (There are more than 16!) LESSON 2.6 Special Angles on Parallel Lines 131 21. Assume the pattern of blue and yellow shaded T’s continues. Copy and complete the table for blue shaded and yellow shaded squares and for the total number of squares. Figure number Number of yellow squares Number of blue squares Total number of squares 1 2 3 5 The T-formation 2 3 4 5 6 … n … 35 … … … … … … LINE DESIGNS Can you use your graphing calculator to make the line design shown at right? You’ll need to recall some algebra. Here are some hints. 1. The x- and y-ranges are set to minimums of |
0 and maximums of 7. 2. The design consists of the graphs of seven lines. 3. The equation for one of the lines is y 1 x 1. 7 4. There’s a simple pattern in the slopes and y-intercepts of the lines. You’re on your own from here. Experiment! Then create a line design of your own and write the equations for it. 132 CHAPTER 2 Reasoning in Geometry ALGEBRA SKILLS 1 ● USING YOUR ALGEBRA SKILLS 1 ● USING YOUR ALGEBRA SKILLS 2 ● USING YO USING YOUR ALGEBRA SKILLS 2 Slope The slope of a line is a measure of its steepness. Measuring slope tells us the steepness of a hill, the pitch of a roof, or the incline of a ramp. On a graph, slope can tell us the rate of change, or speed. y To calculate slope, you find the ratio of the vertical distance to the horizontal distance traveled, sometimes referred to as “rise over run.” e g n a h c l a ic t r ve slope a ge n a h c l t n o z ri ho y2 y1 (x2, y2) (x1, y1) run x1 x2 rise y1 y2 (x2, y1) One way to find slope is to use a slope triangle. Then use the coordinates of its vertices in the formula. x1 x2 x Slope Formula The slope m of a line (or segment) through two points with coordinates x1, y1 is and x2, y2 y y2 m 1 x x 1 2 0. x1 where x2 EXAMPLE Draw the slope triangle and find the slope for AB. y 10 5 B A 5 x 10 Solution Draw the horizontal and vertical sides of the slope triangle below the line. Use them to calculate the side lengths. B (7, 6) y2 Note that if the slope triangle is above the line, you subtract the numbers in reverse order, but still get the same result y2 y1 6 1 5 A (3, 1) (7, 1) x2 x1 7 3 4 USING YOUR ALGEBRA SKILLS 2 Slope 133 ALGEBRA SKILLS 2 ● USING YOUR ALGEBRA SKILLS 2 ● USING YOUR ALGEBRA |
SKILLS 2 ● USING YO The slope is positive when the line goes up from left to right. The slope is negative when the line goes down from left to right. When is the slope 0? What is the slope of a vertical line? EXERCISES In Exercises 1–3 find the slope of the line through the given points. 1. (16, 0) and (12, 8) 2. (3, 4) and (16, 8) 3. (5.3, 8.2) and (0.7, 1.5) 4. A line through points (5, 2) and (2, y) has a slope of 3. Find y. 5. A line through points (x, 2) and (7, 9) has a slope of 7. Find x. 3 6. Find the coordinates of three more points that lie on the line passing through the points (0, 0) and (3, 4). Explain your method. 7. What is the speed, in miles per hour, y represented by Graph A? 8. From Graph B, which in-line skater is faster? How much faster? 9. The grade of a road is its slope, s e l i M 700 500 300 100 given as a percent. For example, a 6 road with a 6% grade has slope. 00 1 It rises 6 feet for every 100 feet of horizontal run. Describe a 100% grade. Do you think you could drive up it? Could you walk up it? Is it possible for a grade to be greater than 100%? 4 6 Hours 2 Graph A y 50 40 30 20 10 s r e t e M x 8 10 Skater 1 Skater 2 x 5 10 15 20 25 Seconds Graph B 10. What’s the slope of the roof on the adobe house? Why might a roof in Connecticut be steeper than a roof in the desert? Adobe house, New Mexico Pitched-roof house, Connecticut 134 CHAPTER 2 Reasoning in Geometry Patterns in Fractals In Lesson 2.1, you discovered patterns and used them to continue number sequences. In most cases, you found each term by applying a rule to the term before it. Such rules are called recursive rules. Some picture patterns are also generated by recursive rules. You find the next picture in the sequence by looking at the picture before it and comparing that to the picture before it, and so on. The Geometer |
’s Sketchpad® can repeat a recursive rule on a figure using a command called Iterate. Using Iterate, you can create the initial stages of fascinating geometric figures called fractals. Fractals have self-similarity, meaning that if you zoom in on a part of the figure, it looks like the whole. A true fractal would need infinitely many applications of the recursive rule. In this exploration, you’ll use Iterate to create the first few stages of a fractal called the Sierpin´ski triangle. In this procedure you will construct a triangle, its interior, and midpoints on its sides. Then you will use Iterate to repeat the process on three outer triangles formed by connecting the midpoints and vertices of the original triangle. This fern frond illustrates self-similarity. Notice how each curled leaf resembles the shape of the entire curled frond. EXPLORATION Patterns in Fractals 135 Activity The Sierpin´ski Triangle Sierpin´ski Triangle Iteration 1. Open a new Sketchpad™ sketch. 2. Use the Segment tool to draw triangle ABC. 3. Select the vertices, and construct the triangle interior. 4. Select AB, BC, and CA, in that order, and construct the midpoints D, E, and F. 5. Select the vertices again, and choose Iterate from the Transform menu. An Iterate dialog box will open. Select points A, D, and F. This maps triangle ABC onto the smaller triangle ADF. 6. In the Iterate dialog box, choose Add A New Map from the Structure pop-up menu. Map triangle ABC onto triangle BED. 7. Repeat Step 6 to map triangle ABC onto triangle CFE. 8. In the Iterate dialog box, choose Final Iteration Only from the Display pop-up menu. 9. In the Iterate dialog box, click Iterate to complete your construction. 10. Click in the center of triangle ABC and hide the interior to see your fractal at Stage 3. 11. Use Shift+plus or Shift+minus to increase or decrease the stage of your fractal. Step 1 Follow the Procedure Note to create the Stage 3 Sierpin´ski triangle. The original triangle ABC is a Stage 0 Sierpin´ski triangle. Practice the last step of the Procedure Note to see how the fractal grows in successive stages. Write a sentence or two explaining what the Sierpin´ski triangle shows you about self- |
similarity. Notice that the fractal’s property of self-similarity does not change as you drag the vertices. Step 2 What happens to the number of shaded triangles in successive stages of the Sierpin´ski triangle? Decrease your construction to Stage 1 and investigate. How many triangles would be shaded in a Stage n Sierpin´ski triangle? Use your construction and look for patterns to complete this table. Stage number Number of triangles 0 1 1 3 2 3 … … n … 50 … What stage is the Sierpin´ski triangle shown on page 135? Step 3 Suppose you start with a Stage 0 triangle (just a plain old triangle) with an area of 1 unit. What would be the shaded area at Stage 1? What happens to the shaded area in successive stages of the Sierpin´ski triangle? Use your construction and look for patterns to complete this table. 136 CHAPTER 2 Reasoning in Geometry Stage number Shaded area … 50 … What would happen to the shaded area if you could infinitely increase the stage number? Step 4 Suppose you start with a Stage 0 triangle with a perimeter of 6 units. At Stage 1 the perimeter would be 9 units (the sum of the perimeters of the three triangles, each half the size of the original triangle). What happens to the perimeter in successive stages of the Sierpin´ski triangle? Complete this table. Stage number Perimeter 0 6 1 9 2 3 … … n … 50 … Step 5 Step 6 What would happen to the perimeter if you could infinitely increase the stage number? Increase your fractal to Stage 3 or 4. If you print three copies of your sketch, you can put the copies together to create a larger triangle one stage greater than your original. How many copies would you need to print in order to create a triangle two stages greater than your original? Print the copies you need and combine them into a poster or a bulletin board display. Sketchpad comes with a sample file of interesting fractals. Explore these fractals and see if you can use Iterate to create them yourself. You can save any of your fractal constructions by selecting the entire construction and then choosing Create New Tool from the Custom Tools menu. When you use your custom tool in the future, the fractal will be created without having to use Iterate. The word fractal was coined by Benoit Mandelbrot (b 1924), a pioneering researcher in this new field of mathematics. He was the first |
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