https://file.aiquickdraw.com/mathbot/file/1725676028100rbfh3d4i.jpg

To solve this problem, let's follow these steps:

1. **Identify given information:**
   - \( MO \) bisects \( \angle PMN \).
   - \( m\angle PMN = 74^\circ \).
   - \( m\angle LOMN = 2x + 7 \).

2. **Understand the implications of the bisector:**
   Since \( MO \) bisects \( \angle PMN \), it divides \( \angle PMN \) into two equal angles. Therefore, each of these angles is half of \( 74^\circ \):

   \[
   m\angle POM = m\angle MON = \frac{74^\circ}{2} = 37^\circ.
   \]

3. **Relate the bisected angle to \( \angle LOMN \):**
   Because \( \angle LOMN \) is the straight angle formed along the line and \( \angle LOMN = 2x + 7 \), and it includes \( \angle MON \), which is \( 37^\circ \):

   \[
   m\angle LOMN = 180^\circ - m\angle MON = 180^\circ - 37^\circ = 143^\circ.
   \]

4. **Set up the equation:**
   Now, equate the expression for \( m\angle LOMN \) to its value:

   \[
   2x + 7 = 143.
   \]

5. **Solve for \( x \):**

   \[
   2x + 7 = 143 \\
   2x = 143 - 7 \\
   2x = 136 \\
   x = \frac{136}{2} = 68.
   \]

So, the value of \( x \) is \( 68 \).

Would you like a detailed explanation of any step, or do you have any questions?

### Related Questions:
1. How do you find the value of angles when given a bisector?
2. What is the significance of a bisector in geometry?
3. How do you solve equations involving angles and linear expressions?
4. Can you explain the relationship between angles on a straight line?
5. What other properties of angles can be used to solve similar problems?

**Tip:** Always double-check your setup and ensure that angles add up correctly in equations involving geometry.

If MO bisects ∠PMN, m∠PMN = 74° and m∠LOMN = [2x + 7]°, find the value of x.

Learn how to find the value of x when given an angle bisector and linear expressions for angles. This problem involves concepts such as the angle bisector theorem and supplementary angles.

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