Let's analyze the geometry problem in the given image step by step.

Key Information from the Diagram:

• The quadrilateral is inscribed in a circle (cyclic quadrilateral).
• We are given the following angle measures:
  • $\angle BDC = 30^\circ$
  • $\angle BCA = 60^\circ$

Objective:

• Find $\angle ABC$.

Using Properties of Cyclic Quadrilaterals:

1. Opposite Angles in a Cyclic Quadrilateral: The sum of opposite angles in a cyclic quadrilateral is always $180^\circ$. This is a key property we will use.

2. Understanding the Given Angles:

   • $\angle BDC = 30^\circ$ (angle subtended by chord $BC$ at point $D$),
   • $\angle BCA = 60^\circ$ (angle subtended by chord $BD$ at point $A$).

3. Applying the Inscribed Angle Theorem: In a circle, the measure of an inscribed angle is half of the measure of the arc it subtends. Here, the arc subtended by $\angle ABC$ is the same as the arc subtended by $\angle BDC$, and similarly for $\angle BCA$.

4. Finding $\angle ABC$: Since the angles $\angle ABC$ and $\angle BDC$ are subtended by the same arc $AC$, we can use the fact that $\angle ABC$ will be double that of $\angle BDC$, making:

   $\angle ABC = 180^\circ - \angle BCA - \angle BDC$

   Substituting the given angles: $\angle ABC = 180^\circ - 60^\circ - 30^\circ = 90^\circ$

Thus, $\angle ABC = 90^\circ$.

Answer:

The correct option is $\boxed{90^\circ}$.

Would you like more details on any of the steps or properties used?

Follow-Up Questions:

1. What is the property of opposite angles in a cyclic quadrilateral?
2. How do we use the inscribed angle theorem to find angles in a circle?
3. What happens to angles subtended by the same arc in a circle?
4. Why is the sum of opposite angles in a cyclic quadrilateral always $180^\circ$?
5. How does the subtended angle change as you move the vertex on the circle?

Tip:

Always remember that in a cyclic quadrilateral, opposite angles are supplementary (i.e., their sum is $180^\circ$). This property is extremely useful in solving problems involving circles.