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To solve for the measure of \( \angle BOC \), we need to analyze the given information:

1. **Congruent Chords \( \overline{AB} \) and \( \overline{BC} \):** Since these chords are congruent, the arcs they subtend (arc \( AB \) and arc \( BC \)) are also congruent.
2. **Central Angle \( \angle AOC = 130^\circ \):** This angle subtends arc \( AC \).

Since arc \( AB \) and arc \( BC \) are congruent, each will measure half of arc \( AC \). Thus:
\[
\text{Measure of arc } AB = \text{Measure of arc } BC = \frac{130^\circ}{2} = 65^\circ.
\]

Now, \( \angle BOC \) is the central angle subtending arc \( BC \), so the measure of \( \angle BOC \) is equal to the measure of arc \( BC \).

Therefore:
\[
\angle BOC = 65^\circ.
\]

The correct answer is:
**B. 65°**

Would you like further explanation or have any questions?

Here are some related questions to consider:
1. What is the relationship between central angles and the arcs they subtend?
2. Why do congruent chords in a circle have congruent arcs?
3. How would the solution change if \( \angle AOC \) were different?
4. What properties of a circle allow us to divide \( \angle AOC \) evenly between arcs \( AB \) and \( BC \)?
5. Can this method be applied if \( AB \) and \( BC \) were not congruent?

**Tip:** Remember, in a circle, central angles are directly proportional to the arcs they subtend.

Given circle O below, if AB and BC are congruent, what is the measure of ∠BOC?

Learn how to find the measure of ∠BOC in a circle with given congruent chords AB and BC, and central angle ∠AOC of 130°. This solution covers key geometry concepts including congruent chords, central angles, and circle properties. Suitable for Grades 9-10.

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