CPM Homework Help : INT2 Problem 12-114

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):
$m\overarc{ABD} = 180º$ because $\overline{AD}$ is a diameter
$m\overarc{AB} + m\overarc{BD} = m\overarc{ABD}$
$106º+m\overarc{BD}=180º$
$m\overarc{BD}=74º$
The measure of the intercepted arc is twice the measure of the inscribed angle
$\begin{array}{l} m\overarc{BD} = 2(x)\\ \phantom{0}\,74° = 2x\\ \phantom{000}\;x=37° \end{array}$
One equation could be $2x = 180º - 106º$

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