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# help

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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

### 1+0 Answers

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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