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To find the value of \(x\) in the given diagram, we can use the properties of cyclic quadrilaterals and the exterior angle theorem.

### Step-by-Step Solution:

1. **Identify the Cyclic Quadrilateral:**
   - The quadrilateral \(KLMN\) is inscribed in the circle, making it a **cyclic quadrilateral**. One key property of cyclic quadrilaterals is that opposite angles sum to \(180^\circ\).
   - Therefore, \( \angle K + \angle M = 180^\circ \) and \( \angle L + \angle N = 180^\circ \).

2. **Use the Exterior Angle Theorem:**
   - The exterior angle theorem for a cyclic quadrilateral states that an exterior angle is equal to the opposite interior angle.
   - Here, \(\angle PNM = 75^\circ\) is an exterior angle, which is equal to the interior opposite angle \(\angle KLM\) (denoted as \(x\)).

3. **Set up the Equation:**
   \[
   x = 75^\circ
   \]

### Conclusion:
The value of \(x\) is \(75^\circ\).

Would you like to explore any other aspects of this problem or any other related concepts?

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### Related Questions:
1. What are the properties of cyclic quadrilaterals?
2. How does the exterior angle theorem apply to different types of quadrilaterals?
3. How can you use the sum of angles in a quadrilateral to solve for unknown angles?
4. What are some common theorems involving angles in a circle?
5. How does the inscribed angle theorem relate to other geometric properties in a circle?

### Tip:
When working with cyclic quadrilaterals, always remember that opposite angles are supplementary, and exterior angles equal the opposite interior angles. These properties are very powerful for solving unknowns.

Learn how to find the angle x in the cyclic quadrilateral KLMN using the exterior angle theorem and properties of cyclic quadrilaterals. Step-by-step solution provided. Suitable for high school students.

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