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Let's analyze the given problem step-by-step:

1. **Understanding the Problem:**
   - \( A, B, C, \) and \( D \) are points on the circumference of the circle with center \( O \).
   - \( BD \) is a diameter of the circle.
   - \( \angle CAB = 48^\circ \).

2. **Properties to Use:**
   - **Angle in a Semi-Circle:** Any angle subtended by a diameter at the circumference is a right angle (\(90^\circ\)).
   - **Angles Subtended by the Same Arc:** Angles subtended by the same arc in the same segment are equal.

### Solutions:

#### (a) Write down the size of angle \( \angle BCD \).

Since \( BD \) is the diameter, \( \angle BCD \) is an angle subtended by the diameter. Therefore, \( \angle BCD = 90^\circ \).

#### (b) Find the size of angle \( \angle BDC \).

By the angle sum property of a triangle in \( \triangle ABC \):
\[
\angle CAB + \angle ABC + \angle BCA = 180^\circ
\]
Given \( \angle CAB = 48^\circ \) and \( \angle BCD = 90^\circ \), the remaining angle \( \angle ABC = 180^\circ - 90^\circ - 48^\circ = 42^\circ \).

#### (c) Find the size of angle \( \angle BOC \).

The angle at the center \( \angle BOC \) is twice the angle at the circumference \( \angle BAC \). Therefore, 
\[
\angle BOC = 2 \times \angle BAC = 2 \times 48^\circ = 96^\circ
\]

#### (d) Find the size of angle \( \angle CAD \).

Since \( \angle CAB \) is given and \( \angle BCD = 90^\circ \), angle \( \angle CAD = \angle BAC = 48^\circ \) (by the property of angles subtended by the same arc in the same segment).

#### (e) Find the size of angle \( \angle COD \).

Since \( BD \) is the diameter, \( \angle COD = 180^\circ \).

#### (f) Find the size of angle \( \angle OCB \).

In \( \triangle OBC \):
- \( \angle BOC = 96^\circ \).
- \( \angle OCB \) and \( \angle OBC \) are the remaining angles.

Since the sum of angles in a triangle is \( 180^\circ \):
\[
\angle OBC = \angle OCB = \frac{180^\circ - 96^\circ}{2} = 42^\circ
\]

Would you like further details or have any questions on this problem?

### Relative Questions:
1. How do you find angles in a cyclic quadrilateral?
2. What is the property of opposite angles in a cyclic quadrilateral?
3. Can you explain the angle subtended by the same arc in a circle?
4. How do you prove that angles in a semi-circle are \( 90^\circ \)?
5. What is the relationship between the central angle and the angle at the circumference?

### Tip:
Always remember that in a circle, the angle at the center is twice the angle at the circumference subtended by the same arc.

A, B, C, and D are points on the circumference of a circle, center O. BD is a diameter of the circle. Angle CAB = 48°. (a) Write down the size of angle BCD. (b) Find the size of angle BDC. (c) Find the size of angle BOC. (d) Find the size of angle CAD. (e) Find the size of angle COD. (f) Find the size of angle OCB.

Learn how to solve for various angles in a circle with diameter BD. This detailed guide covers key circle geometry concepts, such as the angle in a semicircle, the relationship between the central angle and the angle at the circumference, and the angle sum property of triangles. Suitable for students in Grades 9-10.

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