https://file.aiquickdraw.com/mathbot/file/1725260935853wurwrx1s.jpg

To solve the problem, let's analyze the provided diagram and the given conditions:

### Given:
1. \( MNOP \) is a parallelogram.
2. \( NO = NQ \).
3. \( \angle M = 110^\circ \).
4. \( \angle N_2 = 4x \).

### To find:
The value of \( x \).

### Solution:

1. **Understanding the Angles:**
   - Since \( MNOP \) is a parallelogram, opposite angles are equal. Therefore:
     \[
     \angle M = \angle P = 110^\circ
     \]
   - The sum of the interior angles of a parallelogram is \( 360^\circ \), and opposite angles are equal, so:
     \[
     \angle N + \angle O = 360^\circ - 2(110^\circ) = 360^\circ - 220^\circ = 140^\circ
     \]
   - Therefore:
     \[
     \angle N + \angle O = 140^\circ
     \]

2. **Examining \( \angle N \) and \( \angle O \):**
   - The given problem indicates that \( \angle N_2 = 4x \). Since \( NQ = NO \), triangle \( NQO \) is isosceles, and the angles \( \angle N_1 \) and \( \angle O_1 \) are equal. Let's call these angles \( y \).
   - Therefore:
     \[
     \angle N_1 = \angle O_1 = y
     \]
   - Since \( \angle N = \angle N_1 + \angle N_2 \) and \( \angle O = 2y \), we can substitute these values into the sum:
     \[
     y + 4x + 2y = 140^\circ
     \]
     Simplifying:
     \[
     3y + 4x = 140^\circ
     \]

3. **Using the Parallel Lines:**
   - Since \( NO = NQ \), \( \angle N_1 \) and \( \angle O_1 \) are congruent, and by the property of the parallelogram and the isosceles triangle \( NQO \), \( \angle O_1 = \angle O_2 = y \).

4. **Solving for \( x \):**
   - To find \( y \), observe that \( \angle M + \angle N \) must equal \( 180^\circ \) (since they are consecutive angles in a parallelogram):
     \[
     110^\circ + 4x = 180^\circ
     \]
   - Subtracting \( 110^\circ \) from both sides:
     \[
     4x = 70^\circ
     \]
   - Dividing by 4:
     \[
     x = 17.5^\circ
     \]

### Final Answer:
The value of \( x \) is \( 17.5^\circ \).

Do you want more details or have any questions? Here are some related questions to further explore the concepts:

1. What are the properties of angles in a parallelogram?
2. How would the problem change if \( \angle M \) was different?
3. Can we determine the length of the sides if only angles are given?
4. What happens to the values if \( NQ \) is not equal to \( NO \)?
5. How do the properties of isosceles triangles help solve this problem?

**Tip:** Remember, in geometry, always use the properties of shapes to relate angles and sides, as this can often simplify complex problems.

Explore how to solve a geometry problem involving angles in a parallelogram. This detailed solution covers properties of parallelograms, isosceles triangles, and angle relationships. Suitable for high school geometry students.

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