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To solve for the angles \(z\), \(x\), and \(y\) in the given diagram, we can use several geometric properties of circles, specifically related to angles subtended by the same arc and the properties of cyclic quadrilaterals.

### Step 1: Determine Angle \(z\)
Notice that \(\angle BAC\) and \(\angle BDC\) subtend the same arc \(BC\). According to the Inscribed Angle Theorem:
\[
\angle BDC = \angle BAC = 49^\circ
\]
So, \(z = 49^\circ\).

### Step 2: Determine Angle \(x\)
To find \(x\), we can use the fact that the sum of the angles in triangle \(OCD\) (where \(O\) is the center) must be 180°.
Since \(\angle OCD\) and \(\angle ODC\) are given as \(49^\circ\) and \(23^\circ\) respectively, we have:
\[
x = 180^\circ - 49^\circ - 23^\circ = 108^\circ
\]

### Step 3: Determine Angle \(y\)
Now, angle \(y\) can be found using the fact that the angle at the center is twice the angle at the circumference subtended by the same arc. Therefore:
\[
y = \frac{1}{2} \times x = \frac{1}{2} \times 108^\circ = 54^\circ
\]

### Summary
- \(z = 49^\circ\)
- \(x = 108^\circ\)
- \(y = 54^\circ\)

Would you like more detailed explanations or have any questions?

### Follow-up Questions:
1. How do you determine angles in cyclic quadrilaterals?
2. What is the Inscribed Angle Theorem?
3. How do you find the center angle from a chord in a circle?
4. Can the angle at the center be less than twice the angle at the circumference?
5. What are other properties of cyclic quadrilaterals?
6. What if the triangle \(OCD\) was isosceles? How would the angles change?
7. How do central angles relate to the arc length they subtend?
8. Can we solve similar problems using trigonometric identities?

### Tip:
When dealing with circles, always look for relationships between the angles subtended by the same arc or chord. This often leads to straightforward solutions using known theorems.

Learn how to solve angles \(z\), \(x\), and \(y\) in a circle using the Inscribed Angle Theorem and properties of circles. Detailed step-by-step explanation with examples. Suitable for high school students.

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