Geometry: An Interactive Journey to Mastery

Solutions

5. See Figure S.19.4.
   22 + x^2  ^2 gives x 192.


6. Draw one line as shown in Figure S.19.5.
   The inscribed/central angle theorem shows that we have
   congruent alternate interior angles, so the chords are parallel.


7. Draw a chord as shown in Figure S.19.6, and use the
   inscribed/central angle theorem to identify an angle of 20°
   and an angle of 65°.
   Because the angles in a triangle sum to 180°, we have
   ͼ௘ía௘ͽ JLYLQJa = 85°.
   Comment: It is not a coincidence that 85 happens to be the average
   RIDQGWKHWZRRULJLQDODQJOHVPHQWLRQHG&DQ\RXVHHZK\"


8. Draw two chords as shown in Figure S.19.7.
   Using vertical angles and the fact that two inscribed angles from the
   same arc have the same measure, we see that the two triangles formed
   are similar by AA.
   Consequently, matching sides of those triangles come in the same ratio.
   We have acdb. The result follows by cross multiplying.


9. Recall that opposite angles of a parallelogram are congruent.
   Suppose that a parallelogram with two angles of measure x and two
   angles of measure y sits in a circle, as shown in Figure S.19.8.
   Each angle x is half the measure of the arc AB: One arc AB has
   measure 2x, and the other arc AB also has measure 2x. Because the
   two arcs cover the entire circle, 2x + 2x = 360°, giving x = 90°. By
   the same reasoning, y ƒ7KH¿JXUHKDVIRXUULJKWDQJOHVDQGLV
   therefore a rectangle.

A B

x

x

14

(^26)

6

Figure S.19.4

Figure S.19.5

68° 68°

34°

34°

130°

20° 65° 40°

a

A B

C

D

Figure S.19.6

ac

d b

Figure S.19.7

A

B

x

xy

y

Figure S.19.8