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In the given diagram, we are tasked with finding the angle \( \angle DBC \), which is labeled as \( x \).

Let's analyze the diagram and apply the geometric properties:

1. \( O \) is the center of the circle, and \( \angle AOC = 115^\circ \) is the angle subtended by the chord \( AC \) at the center of the circle.
2. \( DBC \) forms a triangle with a tangent \( BC \) to the circle at point \( C \).
3. \( \angle A \) lies between chord \( AD \) and the tangent line \( BC \).

### Step-by-step solution:

1. **Identify the relationship between the central and inscribed angles:**
   - The angle subtended by the same arc at the center of the circle is double the angle subtended at the circumference.
   - \( \angle AOC = 2 \times \angle ADC \). So:
   \[
   \angle ADC = \frac{115^\circ}{2} = 57.5^\circ
   \]

2. **Use the tangent-secant theorem:**
   - The tangent at point \( C \) and the secant \( AD \) create an angle \( \angle A \) at point \( A \). By the tangent-secant theorem, the angle between the tangent and the chord is equal to the angle subtended by the chord on the opposite side of the circle.
   - Therefore, \( \angle DBC = \angle ADC = 57.5^\circ \).

Thus, \( \boxed{x = 57.5^\circ} \).

Would you like further clarification on any of these steps? Here are 5 related questions that expand on this problem:

1. How is the angle subtended at the center related to the angle subtended at the circumference?
2. What is the tangent-secant theorem, and how does it apply to this problem?
3. Can the exterior angle theorem be used to find other angles in this configuration?
4. How would the solution change if the angle at the center was different?
5. How can we calculate the length of the tangent if the radius of the circle is given?

**Tip:** In circle geometry, always look for relationships between angles subtended by the same chord or arc to simplify angle calculations.

Find angle DBC in the given circle geometry problem.

Learn how to find angle DBC using circle geometry principles, including the Central Angle Theorem and Tangent-Secant Theorem. A step-by-step solution guides students through geometric reasoning to solve the problem. Suitable for Grades 9-12.

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