Radii OA, OB, and OC divide the circle into three congruent parts. Let D be a point on the arc AB such that arc AD has half the measure of arc DB. Find the measure of angle OCD.
+9519 OA, OB, OC divide the circle into three congruent parts \(\Longleftrightarrow\) \(\angle AOB = \angle BOC = \angle COA = 120^\circ\)
D is a point on arc AB such that arc AD has half the measure of arc DB \(\Longleftrightarrow\) \(m\stackrel{\mbox{$\large\frown$}}{AD} = 40^\circ\), \(m\stackrel{\mbox{$\large\frown$}}{DB} = 80^\circ\)
\(\text{reflex }\angle COD = m\stackrel{\mbox{$\large\frown$}}{DB} + m\stackrel{\mbox{$\large\frown$}}{BC} = 200^\circ\)
\(m\angle COD = 360^\circ - 200^\circ = 160^\circ\)
Because OC = OD,
\(m\angle OCD = m\angle ODC\)
So \(160^\circ + 2 \left(m\angle OCD\right) = 180^\circ\)
I believe you can continue from here. It's just one step away from the answer.
