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The problem involves a circle with \( \overline{BE} \) as its diameter, and we need to find the measure of \( \angle BCE \).

### Solution:
1. **Understanding the Relationship:**
   - \( \overline{BE} \) is the diameter of circle \( A \), and \( C \) lies on the circle.
   - By the Inscribed Angle Theorem, any angle inscribed in a semicircle (formed by the endpoints of a diameter and a point on the circle) is a right angle (90°).

2. **Application of the Inscribed Angle Theorem:**
   - Since \( \overline{BE} \) is the diameter and \( C \) lies on the circumference, \( \angle BCE \) is inscribed in the semicircle.
   - Therefore, \( \angle BCE = 90^\circ \).

### Final Answer:
The measure of \( \angle BCE \) is \( \boxed{90^\circ} \), so the correct choice is option **3)**.

Would you like further details on the theorem used or have any other questions?

### Related Questions:
1. How does the Inscribed Angle Theorem apply to other points on the circle?
2. What is the difference between a central angle and an inscribed angle?
3. Can the inscribed angle theorem be applied to circles with different diameters?
4. How can you prove that angles inscribed in a semicircle are always 90°?
5. What happens if point \( C \) lies inside the semicircle instead of on the circumference?

### Tip:
Remember, angles inscribed in a semicircle are always right angles, as long as the endpoints of the angle are on the circle's circumference and the third point is anywhere along the semicircle.

BE is the diameter of circle A. What's the measure of ∠BCE?

Explore how to determine the measure of angle ∠BCE in a circle where BE is the diameter. Learn how the Inscribed Angle Theorem is applied to show that any angle inscribed in a semicircle is a right angle. Suitable for Grades 9-10 geometry students.

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