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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.

 Jun 13, 2020
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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.

 Jun 13, 2020

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