https://file.aiquickdraw.com/mathbot/file/1726765060756z8vi0c2p.jpg

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?

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### 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.

In the figure above, ∠BDC = 30° and ∠BCA = 60°. Find ∠ABC.

Find out how to solve for ∠ABC in a cyclic quadrilateral using properties of circles and the inscribed angle theorem. This detailed solution covers key concepts, such as the sum of opposite angles and angles subtended by the same arc. Ideal for students in Grades 9-12.

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