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The problem asks us to find the value of angle \( x \) in the cyclic quadrilateral inscribed in a circle. Here's a step-by-step analysis:

### Key Observations:
1. The figure forms a cyclic quadrilateral \( ABCD \), where \( O \) is the center of the circle.
2. \( \angle ODC = 24^\circ \) (as given in the problem).
3. \( \angle AOB = 82^\circ \) (this is a central angle).
4. We need to determine \( \angle ABC = x \).

### Using the properties of cyclic quadrilaterals:
- Opposite angles of a cyclic quadrilateral sum to \( 180^\circ \). Therefore, \( \angle AOB + \angle ACB = 180^\circ \).
  
Using this property:
\[
\angle AOB = 82^\circ \quad \text{(given)}
\]
Thus, the opposite angle \( \angle ACB \) will be:
\[
\angle ACB = 180^\circ - 82^\circ = 98^\circ
\]

Now, we also know that \( \triangle ODC \) is isosceles, as \( O \) is the center of the circle, making \( OC = OD \). Hence, the angle at \( D \), i.e., \( \angle ODC = 24^\circ \), helps determine the overall geometry.

Finally, using these properties and solving for \( x \), the correct value of \( x \) is \( 98^\circ \), which corresponds to option B.

Would you like further details, or do you have any questions?

### Related Questions:
1. What are the properties of a cyclic quadrilateral?
2. How can central and inscribed angles help solve circle geometry problems?
3. How does an isosceles triangle help simplify circle-related problems?
4. What is the relationship between angles subtended by the same arc?
5. Can we use alternate segment theorem to find angle \( x \) in this case?

#### Tip:
In cyclic quadrilaterals, always remember that the sum of opposite angles is \( 180^\circ \).

In the following figure, O is the center of a circle and ∠ODC = 24°. Find the value of x.

Learn how to find the value of angle x in a cyclic quadrilateral inscribed in a circle with an isosceles triangle. This problem involves using the properties of cyclic quadrilaterals and theorems related to circle geometry. Suitable for students in Grades 9-10.

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