Questions: Circle Relationships: Mastery Test Point O is the center of the circle in the diagram. What is m∠BOA? A. 80° B. 75° C. 70° D. 65°

Solution Steps

• Point O is the center of the circle.
• We need to find the measure of angle ∠BCA.

• Since O is the center, OA and OC are radii of the circle.
• Therefore, triangle OAC is an isosceles triangle with OA = OC.

• In an isosceles triangle, the base angles are equal.
• Let ∠OAC = ∠OCA = x.

• The sum of angles in triangle OAC is 180°.
• Therefore, ∠AOC + ∠OAC + ∠OCA = 180°.
• Given that ∠AOC is a central angle and it is 140° (since the arc it subtends is 140°).

• ∠AOC = 140°.
• So, 140° + x + x = 180°.
• 140° + 2x = 180°.
• 2x = 40°.
• x = 20°.

• ∠BCA is an exterior angle to triangle OAC.
• An exterior angle is equal to the sum of the two non-adjacent interior angles.
• Therefore, ∠BCA = ∠OAC + ∠OCA = 20° + 20° = 40°.

Final Answer