Chapter 8

    1. All odd-numbered angles are equal (1 & 3 are vertical angles, 3 & 5 are zigzag angles, 5 & 7 are vertical, 7 & 9 are zigzags, 9 & 11 are vertical), as are all the even-numbered angles.
  1. We have many pairs of vertical angles, but can’t say if any of the non-vertical angles are equal. (We don’t know, for example, if angles 3 & 5 are equal.)

  1. Yes: they all have angle sum $ 540^{} $ , which can be seen by drawing two non-intersecting diagonals, and thinking about the angle sums in the three triangles produced thereby.

    1. $ 108^{} $ , b) $ 135^{} $ , c) $ 179.64^{} $

  3. No. Each angle would exceed $ 90^{} $ , making the triangle’s angle sum exceed $ 180^{} $ , which can’t happen.

  4. $ ABC $ is determined (by SAS), so we can use trig to get $ AB $ , which equals $ DE $ , since these are vertical angles. Then $ CDE $ will be determined (by ASA), so we can use trig to find $ CE $ . From this, we can get $ FE $ , which is $ CE $ ’s supplement. Then $ DFE $ is determined (by ASA), so trig will get FE for us. Since opposite sides of parallelograms are equal, GH = FE, the length we just obtained.

  5. No, yes, no, yes.

    1. $ ABC DEF $ means that the two triangles are congruent.
  1. $ ABC XYZ $ means that the two triangles are similar.
    1. Yes: by the result proved prior to this exercise (applied over and over), all 10 triangles are similar.
    1. 10 b) 5 c) 15 d) 10 congruent copies, 20 similar copies. e) $ (1 + )/2 $

  3. Let c be the hypotenuse. Then $ c{2}=a{2}+b^{2} $ , so $ c= $ .

  4. Let a be the unknown leg. Then $ a^{2} + l^{2} = c^{2} $ , so $ a^{2} = c^{2} - l^{2} $ . Thus, $ a = $ .

Chapter 9

  1. $ 56^{} $ , 2.97, 3.58; $ 39.8^{} $ , 6.15, 5.12; $ 61^{} $ , 2.22, 4.57.

  2. $ $ , $ 51.3^{} $ , $ 38.7^{} $ ; $ 4 $ , $ 51.1^{} $ , $ 38.9^{} $ ; $ $ , $ 35.3^{} $ , $ 54.7^{} $ .

4.75.7° 7.26’6’’ 10.(see the answers to Exercise 1) 11.36.9°

  1. The values of cosine at these angles are the values of sine at their complementary angles. For example, there’s no need to memorize cos $ 60^{} $ ; we know it equals sin $ 30^{} $ , which is $ $ (a fact we have already memorized).

    1. $ (70 - ) $ 17. $ ^{} = $ , $ ^{} = 1 $ , $ ^{} = $ . 18. a) $ ^{} $ , b) $ $ , c) 1

  3. See the name of Arthur Conan Doyle’s second Holmes novel. 22. $ = $ .

  4. b, e, h are FALSE. The rest are true. 27. a) $ $ , b) $ $ , c) $ $ , d) 0

  5. See the answers to Exercise 1. (These are the same triangles, but you can now solve them more efficiently.)

  6. $ AF $ , $ FE $ , AE = 3.60, $ FE = 33^{} $ , $ AE = 57^{} $

\[ AF=EF\approx1.66,\quad AE\approx2.35,\quad F\hat{AE}=E\hat{AF}=45^{\circ} \]

  1. Earth’s volume is $ 4^{3} = 64 $ times that of the moon. 35. Approximately $ 35.3^{} $ 36. 0.236 units.

    1. 3 miles, 80 feet; 2 miles, 3972 feet. c) 16’6’‘, 66’0’’.
  1. The distance from his eyes is greater, but only by about a quarter of an inch.

  1. Approximately 15’5’’.

  2. $ ^{} = $ , $ ^{} = $ . All the other values can be obtained from these.