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To find the measure of \( \angle ZXY \), we can use the properties of a circle and the relationship between the lengths of the arcs and angles.

### Steps to Solve:

1. **Identify Given Information**:
   - The arc \( ZY \) is labeled as \( 6x + 16 \).
   - The arc \( ZX \) is labeled as \( 13x - 7 \).

2. **Recognize the Angle's Relationship**:
   - \( \angle ZXY \) is the angle subtended by arc \( ZY \) at point \( X \) on the circumference.
   - The measure of \( \angle ZXY \) is half the measure of arc \( ZY \) (due to the Inscribed Angle Theorem).

3. **Find the Measure of Arc \( ZY \)**:
   The problem doesn't provide the measure directly but gives an expression \( 6x + 16 \). However, we can't directly determine \( x \) without additional information (e.g., the total arc or a specific value for another angle or arc).

4. **Set Up an Equation** (if assuming the circle is complete and these are the only arcs given, or they could add up to the circle's total arc):
   - Typically, the entire circle sums to \( 360^\circ \), but since we have expressions for two arcs, the sum of these could equal the remaining arc, depending on context.

Assuming no other arcs are provided:
\[ \text{Arc ZY} + \text{Arc ZX} = 360^\circ \]
\[ (6x + 16) + (13x - 7) = 360 \]
\[ 19x + 9 = 360 \]
\[ 19x = 351 \]
\[ x = \frac{351}{19} = 18.47^\circ \text{ (approximately)} \]

5. **Calculate \( \angle ZXY \)**:
   Now, using \( x \approx 18.47 \) in arc \( ZY \):
\[ \text{Arc ZY} = 6(18.47) + 16 \approx 111.82^\circ \]
\[ \text{Measure of } \angle ZXY = \frac{1}{2} \times 111.82^\circ \approx 55.91^\circ \]

Thus, \( m\angle ZXY \approx 55.91^\circ \).

Would you like more details or have any other questions?

### Related Questions:
1. What is the measure of the other angle \( \angle ZYX \) in the triangle?
2. How would the angle \( \angle ZXY \) change if the arc \( ZY \) was smaller?
3. Can you find the remaining arc \( XY \) using the given values of \( x \)?
4. How does the sum of the interior angles of the triangle relate to the circle?
5. What other circle theorems are relevant to finding angles inside a circle?

### Tip:
Remember that the measure of an inscribed angle is always half the measure of the arc it subtends. This is crucial for solving problems related to circles and angles.

Learn how to find the measure of angle ZXY using arc measures and the Inscribed Angle Theorem in circle geometry. This problem involves calculating angles subtended by arcs on a circle. Suitable for high school students studying circle theorems.

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