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In this problem:

- We have a circle with center \( O \), circumscribing triangle \( ABC \).
- Line \( CD \) is perpendicular to chord \( AB \), indicating that \( O \) lies on the perpendicular bisector of \( AB \) since it’s the center of the circle.
- We are given that \( \angle OAC = 32^\circ \) and need to find the values of \( x \), \( y \), and \( z \) as marked in the diagram.

### Step-by-Step Solution

1. **Identifying \( \angle OCA \):**
   Since \( O \) is the center of the circle, \( OA = OC \), making \( \triangle OAC \) an isosceles triangle. Therefore,
   \[
   \angle OCA = \angle OAC = 32^\circ.
   \]

2. **Finding \( \angle AOC \):**
   In \( \triangle OAC \), the sum of angles is \( 180^\circ \):
   \[
   \angle AOC + \angle OAC + \angle OCA = 180^\circ.
   \]
   Substituting the values we know:
   \[
   \angle AOC + 32^\circ + 32^\circ = 180^\circ,
   \]
   \[
   \angle AOC = 180^\circ - 64^\circ = 116^\circ.
   \]
   Therefore, \( z = 116^\circ \).

3. **Finding \( \angle ABC = y \):**
   Since \( \angle AOC \) is an angle at the center subtending arc \( AC \), the angle subtended by the same arc at the circumference, \( \angle ABC \), will be half of \( \angle AOC \) (by the angle subtended by the same arc theorem). Thus:
   \[
   \angle ABC = \frac{1}{2} \times 116^\circ = 58^\circ.
   \]
   So, \( y = 58^\circ \).

4. **Finding \( \angle DAB = x \):**
   Since \( CD \) is perpendicular to \( AB \), angle \( \angle DAB \) is half of \( \angle ACB \) (because it's an angle formed by the chord \( AB \) and the diameter, applying the perpendicular bisector property).
   
   Using the same logic of the angle subtended by the arc, we conclude that:
  

In the given diagram, O is the centre of the circle circumscribing the triangle ABC; CD is perpendicular to chord AB. \( \angle OAC = 32^\circ \). Find each of the unknown angles x, y, and z.

Solve for unknown angles in a circle given the center, a perpendicular chord, and one known angle. This problem applies circle theorems such as angle subtended by the same arc and properties of an isosceles triangle. Suitable for Grades 9-10 students.

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