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 = 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.