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Let's go through each part of the problem:

### Given Information
- **AC** is a diameter of the circle **D**.
- **mBC** (arc BC) = 70°.

Since **AC** is a diameter, **angle ABC** is a right angle (90°) by the Inscribed Angle Theorem (angle subtended by a semicircle is a right angle).

### Required Measures
Let's calculate each measure step by step:

1. **m∠ABC**
   - Since **AC** is the diameter, **∠ABC** is 90°.
   - **Answer**: \( m∠ABC = 90° \).

2. **m∠BDC**
   - **∠BDC** is an inscribed angle subtending arc **BC**. The measure of an inscribed angle is half the measure of its intercepted arc.
   - Given **mBC** = 70°, then \( m∠BDC = \frac{70°}{2} = 35° \).
   - **Answer**: \( m∠BDC = 35° \).

3. **m∠CAB**
   - **∠CAB** is an inscribed angle subtending the semicircle **AC**. Since **AC** is the diameter, **∠CAB** is also a right angle.
   - **Answer**: \( m∠CAB = 90° \).

4. **m∠ABC** (already calculated in part 1)
   - **Answer**: \( m∠ABC = 90° \).

5. **m∠BCA**
   - In triangle **ABC**, the sum of angles must be 180°. We already know two angles: \( m∠ABC = 90° \) and \( m∠CAB = 90° \).
   - Thus, \( m∠BCA = 180° - (m∠ABC + m∠CAB) = 180° - (90° + 35°) = 55° \).
   - **Answer**: \( m∠BCA = 55° \).

### Summary of Answers
- \( m∠ABC = 90° \)
- \( m∠BDC = 35° \)
- \( m∠CAB = 90° \)
- \( m∠ABC = 90° \) (repeated for completeness)
- \( m∠BCA = 55° \)

Would you like more details or have any questions?

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### Related Questions
1. What is the Inscribed Angle Theorem?
2. How does the diameter of a circle affect inscribed angles?
3. Can an angle subtending a semicircle ever be more than 90°?
4. How are arc measures related to inscribed angles in a circle?
5. What other properties are unique to diameters in circles?

### Tip
Remember that angles subtended by the same arc are always equal. This can simplify many problems involving circles and inscribed angles.

Given that AC is a diameter of circle D and mBC = 70°, find each measure:
6. m∠ABC
7. m∠BDC
8. m∠CAB
9. m∠ABC
10. m∠BCA.

Learn how to determine the angle measures in a circle where AC is a diameter and mBC = 70°. Using the Inscribed Angle Theorem and the concept of right angles in circles, this solution covers step-by-step calculations for m∠ABC, m∠BDC, m∠CAB, and m∠BCA.

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