Let's analyze the problem:

We are given a circle with:

1. $\angle ABC = 4x$
2. $\angle ADC = 11x$
3. $\angle BCD = 98^\circ$

We are tasked to determine:

1. The value of $x$,
2. $\angle ABC$,
3. $\angle ADC$, and
4. $\angle BAD$.

Step 1: Use the angle properties of a circle.

In a circle:

• Opposite angles subtending the same arc are supplementary. Hence: $\angle ABC + \angle ADC = 180^\circ$ Substituting the given expressions: $4x + 11x = 180^\circ$ $15x = 180^\circ$ $x = 12^\circ$

Step 2: Calculate $\angle ABC$ and $\angle ADC$.

• $\angle ABC = 4x = 4(12^\circ) = 48^\circ$
• $\angle ADC = 11x = 11(12^\circ) = 132^\circ$

Step 3: Determine $\angle BAD$.

The angle $\angle BAD$ is subtended by the same arc as $\angle BCD$. Therefore: $\angle BAD = \angle BCD = 98^\circ$

Final Answer:

1. $x = 12^\circ$,
2. $\angle ABC = 48^\circ$,
3. $\angle ADC = 132^\circ$,
4. $\angle BAD = 98^\circ$.

Let me know if you'd like a detailed explanation for any step or have further questions!

Related Questions:

1. Why do opposite angles subtending the same arc add up to $180^\circ$?
2. What are the properties of cyclic quadrilaterals in circles?
3. How can you prove that $\angle BAD = \angle BCD$?
4. What happens if one of the given angles changes? How would you calculate the others?
5. How does this relate to the inscribed angle theorem?

Tip:

In problems involving circles, always consider the properties of arcs, inscribed angles, and supplementary angles to simplify your calculations.