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To solve for angle \( r \) in the circle centered at \( O \), we can use the following geometric properties of circles and angles:

1. **Isosceles Triangle**: Since \( OX \) and \( OZ \) are radii of the circle, the triangle \( OXZ \) is isosceles, meaning that the angles opposite these sides (i.e., \( \angle XZO \) and \( \angle ZOX \)) are equal.

2. **Angle at the Center**: The angle \( \angle YOZ \), subtended by arc \( YZ \) at the center, is twice the angle subtended by the same arc at the circumference, \( \angle XYZ \). This is the "Angle at the Center Theorem."

Using these ideas, let’s proceed with the calculations:

### Step-by-Step Solution:
1. **Using the Angle at the Center Theorem**:
   According to the Angle at the Center Theorem, the angle subtended at the center (i.e., \( \angle YOZ \)) is twice the angle subtended at the circumference (i.e., \( \angle XYZ = 64^\circ \)).

   Therefore:
   \[
   \angle YOZ = 2 \times 64^\circ = 128^\circ
   \]

2. **Sum of Angles in Triangle \( OXZ \)**:
   The sum of angles in a triangle is \( 180^\circ \). In triangle \( OXZ \), we know the angle at the center, \( \angle YOZ = 128^\circ \), and the remaining two angles are \( r \) and \( r \) because \( OX = OZ \) (isosceles triangle).

   Thus, we have:
   \[
   2r + 128^\circ = 180^\circ
   \]
   Solving for \( r \):
   \[
   2r = 180^\circ - 128^\circ = 52^\circ
   \]
   \[
   r = \frac{52^\circ}{2} = 26^\circ
   \]

Thus, the size of angle \( r \) is \( 26^\circ \).

### Final Answer:
\[
r = 26^\circ
\]

Would you like further details or clarifications?

Here are 5 related questions you might find useful:
1. What is the Angle at the Center Theorem, and why is it significant in circle geometry?
2. How do you determine if a triangle inside a circle is isosceles?
3. What other geometric theorems are commonly used when working with angles in circles?
4. If one of the radii in the triangle was extended, how would it affect the angles?
5. Can the angle \( r \) be calculated using any other method, such as trigonometry?

### Tip:
Always remember to check if a triangle formed inside a circle includes two radii. This will often help you recognize an isosceles triangle, simplifying angle calculations.

The circle is centered at O. Calculate the size of angle r. Fully justify your answer.

Solve the circle geometry problem where angle r is calculated using the Angle at the Center Theorem. Understand key concepts such as isosceles triangles and the sum of angles in a triangle. Ideal for students in Grades 9-12.

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