Let's analyze and solve the problem provided in the image.

The given problem involves a cyclic quadrilateral inscribed in a circle, with the following details:

1. $\angle ABC = 4x$, $\angle ADC = 11x$, and $\angle BCD = 98^\circ$.
2. We are tasked to:
   • Find the value of $x$,
   • Determine $\angle ABC$,
   • Determine $\angle ADC$,
   • Find $\angle BAD$.

Step 1: Key Property of a Cyclic Quadrilateral

The sum of opposite angles of a cyclic quadrilateral is $180^\circ$. Hence, for the quadrilateral $ABCD$: $\angle ABC + \angle ADC = 180^\circ$ Substitute the given values: $4x + 11x = 180$ $15x = 180$ $x = 12$

Step 2: Calculate Each Angle

Now substitute $x = 12$ into the expressions for the angles:

1. $\angle ABC = 4x = 4(12) = 48^\circ$,
2. $\angle ADC = 11x = 11(12) = 132^\circ$,
3. $\angle BAD$: Since $\angle BAD$ subtends the same arc as $\angle BCD$, by the Inscribed Angle Theorem: $\angle BAD = \angle BCD = 98^\circ$

Final Answers:

• $x = 12$,
• $\angle ABC = 48^\circ$,
• $\angle ADC = 132^\circ$,
• $\angle BAD = 98^\circ$.

Would you like further clarifications or a breakdown of the concepts used here?

Related Questions:

1. How does the property of cyclic quadrilaterals help solve angle-related problems?
2. What is the Inscribed Angle Theorem, and how is it applied?
3. How can you check if a quadrilateral is cyclic?
4. What happens to the opposite angles if a cyclic quadrilateral degenerates into a triangle?
5. Can this problem be extended to circles with multiple inscribed shapes?

Tip:

For any cyclic quadrilateral, always remember: $\text{Opposite angles sum up to } 180^\circ.$ It is a powerful property for solving geometry problems involving circles.