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 $180^\circ$.