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To find the measure of angle \( \angle OBC \), we can use the given information in the diagram along with properties of circles and angles in cyclic quadrilaterals.

### Step 1: Analyze the given angles
- \( \angle BAD = 28^\circ \)
- \( \angle CAD = 24^\circ \)

### Step 2: Find \( \angle BAC \)
Using the angles given at \( A \):
\[
\angle BAC = \angle BAD + \angle CAD = 28^\circ + 24^\circ = 52^\circ
\]

### Step 3: Use the property of a cyclic quadrilateral
Since \( ABCD \) is a cyclic quadrilateral (all points lie on a circle), the opposite angles of a cyclic quadrilateral sum to \( 180^\circ \). However, here we are more interested in triangle \( OBC \) where \( O \) is the center of the circle.

### Step 4: Central angle and inscribed angle relationship
The angle subtended by a chord at the center of a circle (which is \( \angle BOC \)) is twice the angle subtended at any point on the circumference (which is \( \angle BAC \)).

Thus:
\[
\angle BOC = 2 \times \angle BAC = 2 \times 52^\circ = 104^\circ
\]

### Step 5: Use triangle sum property in \( \triangle OBC \)
In triangle \( OBC \), the sum of the angles is \( 180^\circ \). Let \( \angle OBC = x \).

\[
x + \angle OCB + \angle BOC = 180^\circ
\]
Since \( OB = OC \) (radii of the circle), \( \angle OBC = \angle OCB \). Let this common angle be \( x \).

\[
x + x + 104^\circ = 180^\circ
\]
\[
2x = 76^\circ
\]
\[
x = 38^\circ
\]

Thus, \( \angle OBC = 38^\circ \).

### Conclusion
The measure of \( \angle OBC \) is \( 38^\circ \).

Would you like further details or have any questions about this solution?

Here are 5 related questions that expand on the concepts used:

1. What is the measure of \( \angle OCB \) in the same diagram?
2. Can you prove that \( OB = OC \) using triangle congruence?
3. If \( \angle BOD \) is required, how would you calculate it?
4. How does the property of cyclic quadrilaterals help in solving this problem?
5. What would \( \angle OBC \) be if \( \angle BAC \) were \( 60^\circ \)?

**Tip:** In circle geometry, always consider using the relationship between central angles and inscribed angles as it often simplifies the process of finding unknown angles.

Learn how to find the measure of angle OBC in a cyclic quadrilateral using circle geometry principles. Step-by-step solution includes analyzing given angles, applying properties of cyclic quadrilaterals, and using triangle sum property. Suitable for high school students studying geometry.

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