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Chapter 7, Lesson 1 85 Chapter 7, Lesson 1 85 Chapter 7, Lesson 1 18. They seem to be equal (and they seem to bisect each other). Set I (pages 260-261) Rhombuses. The aerial photographer Georg Gerster, the 19. They seem to be equal (and the opposite author of Below from Above (Abbeville Press, sides seem to be parallel). 1986), wrote about his photograph of the freeway overpass covered by carpets: 20. They seem to be perpendicular (and they "Years after having taken this photograph, I am seem to bisect each other). still amazed at this super-scale Paul Klee .... In Europe, special mats, a Swiss development, have •21. They are equiangular (or each of their been adopted. They retain the heat generated as angles is a right angle). the concrete hardens and make it unnecessary to 22. They are equilateral. keep wetting the concrete. The new procedure is more economical, though less colorful " Parallelograms. Theorem 24. 23. They seem to be parallel and equal. • 1. Two points determine a line. 24. They seem to be equal. 2. The sum of the angles of a triangle is 180°. 25. They seem to bisect each other. •3. Addition. SAT Problem. 4. Betweenness of Rays Theorem. •26. 2x + 2y = 180. (2x + 2y + 180 = 360.) 5. Substitution. 27. x + y = 90. Corollary. Set 11(pages 262-264) •6. The sum of the angles of a quadrilateral is Martin Gardner dedicated his book of mathemati• 360°. cal recreations, Penrose Tiles to Trapdoor Ciphers 7. Division. (W. H. Freeman and Company, 1989) to Roger Penrose with these words: 8. An angle whose measure is 90° is a right "To Roger Penrose, for his beautiful, surprising angle. discoveries in mathematics, physics, and •9. A quadrilateral each of whose angles is a cosmology; for his deep creative insights into right angle is a rectangle. how the universe operates; and for his humility in not supposing that he is exploring only the • 10. Each angle of a rectangle is a right angle. products of human minds." 11. All right angles are equal. Two chapters of Gardner's book are about Carpets. Penrose tilings, a subject full of rich mathematical ideas. The two tiles in the exercises come from a 12. Rectangles. rhombus with angles of 72° and 108°. John Norton Conway calls them "kites" and "darts." 13. Right. Amazingly, in all nonperiodic tilings of these tiles, •14. Lines that form right angles are perpen• the ratio of the number of kites to the number of dicular. darts is the golden ratio: = 1.618 ...! There 15. In a plane, two lines perpendicular to a third line are parallel. are a number of simple ways to prove that there are infinitely many ways to tile the plane with Rectangles. these two tiles. Even so, any finite region of one tiling appears infinitely many times in every other 16. They are equal and they are right angles. tiling; so no finite sample of a Penrose tiling can • 17. They seem to be parallel and equal. be used to determine which of the infinitely 86 Chapter 7, Lesson 1 nnany different tilings it is! Students interested in •40. Six sides, three diagorials, and four triangles. such mind-boggling ideas will enjoy reading Gardner's material and doing the experiments 41. Example figure: suggested in it. The exercises on the angles of polygons are intended to encourage students to realize that memorizing a result such as "each angle of an (n-2)i8o°„ equiangular n-gon has a measure of n is not important. What is important is for the student to feel comfortable and confident in being able to reconstruct such a result if it is 42. Eight sides, five diagonals, and six triangles. needed. •43. n-3. Penrose Tiles. 44. n-2. 28. The one labeled ABCD. 45. A quadrilateral has 4 - 3 = 1 diagonal which •29. 144°. (360°-3-72°.) forms 4-2 = 2 triangles. 30. •46. 720°. (4 X 180°.) 720° •47. 120°. (^.) 6 48. 1,080°. (6 X 180°.) 1 080° 49. 135°. (^^^.) • 31. SSS. 8 32. All are isosceles. 51. •50. (n-2)180°n . 33. They seem to be collinear. Linkage(n-2)180 Problems.° 34. They are collinear because •52. Their sum is 360°. ZACE = 72° + 108° = 180°. 53. Their sum is 180° if we think of the linkage 35. Yes. It appears that it could be folded along as a triangle. (It is 360° if we think of the the line through A, C, and E so that the two linkage as a quadrilateral, because halves would coincide. ZBCD = 180°.) 36. No. Quadrilateral ABED is equilateral but •54. They all appear to be smaller. not equiangular. 55. ZA + ZB + ZAPB = 180°, and Gentstone Pattern. ZC + ZD + ZCPD = 180°; so •37. Triangles, quadrilaterals, and an octagon. ZA + ZB + ZAPB = ZC + ZD + ZCPD. Because ZAPB = ZCPD, it follows that 38. Five sides, two diagonals, and three ZA + ZB = ZC + ZD by subtraction. triangles. Quadrilateral Angle Sum. 39. Example figure: 56. Addition. 57. Substitution. 58. Subtraction. Chapter 7, Lesson 2 87 59. Zl and Z2 are a linear pair; so they are 6. BC looks longer than CF, but they are supplementary and their sum is 180°. actually the same length. Likewise for Z3 and Z4. 7. The opposite angles of a parallelogram are (Zl + Z2) + (Z3 + Z4) = S; so equal. 180° + 180° = S, and so S = 360°. •8. An exterior angle of a triangle is greater Set III (page 264) than either remote interior angle. This triangle-to-square puzzle is the puzzle of 9. Substitution. Henry Dudeney first introduced in the Set III exercise of Chapter 1, Lesson 3. It is perhaps Theorem 26. surprising that, by carefully comparing the two 10. The opposite sides of a parallelogram are figures and knowing that the angles of an equal. equilateral triangle and a square are 60° and 90° respectively, we can find all of the remaining • 11. The opposite sides of a parallelogram are angles from just the one given. (By using the fact parallel. that the triangle and square have equal areas and 12. Parallel lines form equal alternate interior applying some simple trigonometry to the angles. triangle labeled D, we can find the exact values of all of the angles in the figure. The 41° angle, 13. ASA. 4/7 14. Corresponding parts of congruent triangles for example, is rounded from Arcsin —.) are equal. • 15. A line segment is bisected if it is divided into two equal segments. Point Symmetry. 16. •17. The diagonals of a parallelogram bisect each other. Chapter 7, Lesson 2 • 18. Two points are sjmimetric with respect to a point if it is the midpoint of the line segment Set I (pages 267-268) connecting them. 1. The first card (the card with the picture of 19. Parallel lines form equal alternate interior the baseball player). angles. •2. Point. 20. Vertical angles are equal. •3. See if the figure looks exactly the same 21. ASA. when it is turned upside down. 22. Corresponding parts of congruent triangles Optical Illusion. are equal. •4. Three. 23. Two points are symmetric with respect to a •5. Yes. The opposite sides of a parallelogram point if it is the midpoint of the line segment are equal. connecting them. 88 Chapter 7, Lesson 2 Set 11(pages 268-269) 34. Point E is the midpoint of AD. In encyclopedias and unabridged dictionaries, the •35. AD = 2DC. entry for "parallelogram" is followed by an entry for "parallelogram law." The Encyclopedia 36. They are supplementary. Parallel lines form Britannica describes it as the "rule for obtaining supplementary interior angles on the same geometrically the sum of two vectors by side of a transversal. constructing a parallelogram. A diagonal will give •37. 180. the requisite vector sum of two adjacent sides. The law is commonly used in physics to determine 38. 90. a resultant force or stress acting on a structure." The except from Newton's Mathematical Principles 39. Isosceles. of Natural Philosophy \s taken from the original 40. Right. Because x + y = 90, ZBEC = 90°. Latin edition published in 1687. It immediately follows statements of his three laws of motion. 41. BEIEC. A specific measure for ZBAD in the figure for Two Parallelograms. exercises 47 through 50 was given to make the exercises a little easier. Actually, an/measure other than 60° or 120° will work. A nice problem for better students would be to rework the exercises letting ZBAD = x°and to explain what happens when x = 60 or 120. Parallelogram Rule. 24. Corollary 43. Because x, y, and z are the measures of the angles of a triangle, x + y + z = 180. Therefore, ZABC is a straight angle; so A, B, and C are collinear. 44. Because ABDE and BCDE are parallelograms, AB II ED and BC II ED. If A, B, and C are not C D collinear, then through B there are two lines parallel to ED. This contradicts the Parallel •26. 10. (5 cm represents 50 lb; so 1 cm represents Postulate; so A, B, and C must be collinear. 10 lb.) •45. All three (AABE s ADEB = ABCD). 27. 25.(2.5x10 = 25.) 46. No. 28. About 6 cm. Hidden Triangles. 29. About 60. (6x10 = 60.) •30. About 24°. Angle Bisectors. 31. •32. ZAEB = :c°; ZCED = y° 48. AADE, ACDF, and AEBF. 33. AB,AE,andED. 49. SAS. Chapter 7, Lesson 3 89 50. Three. In addition to AABE and ABCF, ADEF Pop-Up Parallelogram. is equilateral because its sides are the • 1. A quadrilateral is a parallelogram if its corresponding sides of the three congruent triangles.
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