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This problem involves a cyclic quadrilateral, as the four vertices of the quadrilateral lie on a circle. In a cyclic quadrilateral, opposite angles are supplementary, meaning their sum is 180°.

From the figure, we know:
- One angle is 93° (y).
- Another angle is 58° (which appears to be opposite to angle \( z \)).

### Step 1: Find angle \( z \)
Since \( z \) is opposite to the angle of 58°, and opposite angles in a cyclic quadrilateral are supplementary:
\[
z + 58^\circ = 180^\circ
\]
\[
z = 180^\circ - 58^\circ = 122^\circ
\]

### Step 2: Find angle \( x \)
Angle \( x \) is opposite to the given angle \( y \) of 93°, and these angles should also be supplementary:
\[
x + 93^\circ = 180^\circ
\]
\[
x = 180^\circ - 93^\circ = 87^\circ
\]

### Step 3: Verify angle \( y \)
We already know that \( y = 93^\circ \), so this is consistent.

### Final Answer:
The measures of the angles are:
- \( x = 87^\circ \)
- \( y = 93^\circ \)
- \( z = 122^\circ \)

However, none of the given choices (A, B, C, D) match this exactly, so there may be a different approach needed or a typo in the options. 

Do you want more details, or have any questions about cyclic quadrilaterals or supplementary angles?

Here are five related questions:
1. How are cyclic quadrilaterals defined geometrically?
2. Can opposite angles in any quadrilateral be supplementary, or only in cyclic ones?
3. What are other properties of angles inside a circle involving chords and arcs?
4. How would the problem change if this were not a cyclic quadrilateral?
5. Can the angles of any quadrilateral sum to something other than 360°?

**Tip**: When dealing with figures involving circles, remember key theorems about angles and arcs—such as the fact that opposite angles in cyclic quadrilaterals sum to 180°.

What are the measures of angles x, y, and z in the figure below?

Learn how to calculate the measures of angles x, y, and z in a cyclic quadrilateral. This problem involves understanding cyclic quadrilateral properties, where opposite angles sum to 180°. Suitable for students in Grades 9-12.

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