IN THE FIGURE , O IS THE CENTER OF THE CIRCLE AND SEGMENT BC IS CONGRUENT TO SEGMENT OA FIND FIND EACH MEASURE

"If segments BC and OA are congruent, then segments BC and OB are also congruent since OA and OB are radii.  The radii of a circle have equal lengths.

Therefore, BC ≅ OA ≅ OB  and these segments form an equilateral isosceles triangle with 60°-60°-60° angles.

The following are measures derived from statements above:

m∠BOC = 60°  (Central angle)

m∠BOC and m∠COA are supplementary.

m∠COA = 120°

Half circle = 180°

1) arc BC = 60°

Proof: The measure of central angle and its intercepted arc are congruent.

2) arc BDA = 180°

Proof: Half circle = 360°/2  or  180°

3) arc BAC = 300°

Proof:

arc BAC = 360° - arc BC  ⇒  arc BAC = 360° -60° = 300°

4) arc AC = 120°

Proof:

arc AC = 180° - arc BC   ⇒  arc AC = 180° - 60° = 120°

5) arc CBA = 240°

Proof:

arc CBA = 360° - arc AC    ⇒   arc CBA = 360° - 120° = 240°"

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