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The question in the image asks to calculate angle ∠BCD and provide reasons for the answer based on the diagram where:

- \( A \), \( B \), and \( C \) lie on a circle with diameter \( AC \).
- \( \angle BAC = 63^\circ \).
- The line \( AC \) is extended to a point \( D \).

### Step-by-step solution:

1. **Recognize the geometric properties:**
   - \( AC \) is the diameter of the circle, and point \( B \) lies on the circumference.
   - This means triangle \( ABC \) is a right triangle according to the **Thales Theorem** (since \( \angle ABC \) subtends a semicircle).

   Therefore, \( \angle ABC = 90^\circ \).

2. **Find \( \angle BCA \):**
   In triangle \( ABC \), the angles sum to \( 180^\circ \). Using the known angles \( \angle ABC = 90^\circ \) and \( \angle BAC = 63^\circ \):
   \[
   \angle BCA = 180^\circ - 90^\circ - 63^\circ = 27^\circ
   \]

3. **Determine \( \angle BCD \):**
   Since \( D \) is an external point and the line \( AC \) is extended, the angle \( \angle BCD \) is an **exterior angle** to triangle \( ABC \).

   The exterior angle theorem tells us that an exterior angle is equal to the sum of the two opposite interior angles of the triangle. Therefore:
   \[
   \angle BCD = \angle BAC + \angle BCA = 63^\circ + 27^\circ = 90^\circ
   \]

### Conclusion:
\[
\boxed{\angle BCD = 90^\circ}
\]
The reason is because \( \angle BCD \) is the exterior angle of triangle \( ABC \), and it equals the sum of the two opposite interior angles.

Would you like further details or explanations on any step?

Here are 5 related questions:
1. How does the Thales Theorem apply to different circles and triangles?
2. Can you prove the exterior angle theorem geometrically?
3. What are the properties of angles in a semicircle?
4. How can the sum of angles in any triangle be proven to be 180 degrees?
5. How does extending a line influence the properties of exterior angles?

**Tip:** Always look for right triangles when dealing with diameters in circle geometry; they often simplify problems!

A, B, and C lie on a circle with diameter AC. AC is extended to D, and angle BAC = 63°. Work out angle BCD and give reasons to explain your answer.

Learn how to calculate angle BCD when given that A, B, and C lie on a circle with diameter AC and angle BAC = 63°. This problem uses Thales Theorem and the Exterior Angle Theorem. Suitable for students in Grade 9-10.

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