CPM Homework Help : GC Problem 11-10

For each of the relationships represented in the diagrams below, write and solve an equation for $x$ and/or $y$ . Justify your method. In part (a), assume that $C$ is the center of the circle.

Solution (a):
$m\overarc{ABD}=180°$ because $\overline{AD}$ is a diameter
$m\overarc{AB} + m\overarc{BD} = m\overarc{ABD}$
$106º+m\overarc{AB}=180°$
$m\overarc{AB}=74°$
The measure of the intercepted arc is twice the measure of the inscribed angle
$\begin{array}{l} m\overarc{AB} = 2(x)\\ \phantom{000}74° = 2x\\ \phantom{0000}x=37° \end{array}$
One equation could be $2x=180°-106°$

Hint (b):
Which angles are supplementary? This will help you write equations to solve for $x$ and $y$ .

Hint (c):
Use the Law of Sines.
Answer (c):
$x = 7.9$ , $y = 7.2$

Hint (d):
Use the Triangle Angle Sum Theorem.