Let's analyze the given image and solve for $m \angle MPQ$.

Key Observations:

• The diagram shows two intersecting lines and the measure of two angles:
  • $\angle MPQ = (4x + 12)^\circ$
  • $\angle NQP = (5x - 18)^\circ$

These two angles, $\angle MPQ$ and $\angle NQP$, are vertical angles, meaning they are equal in measure.

Step-by-Step Solution:

1. Since vertical angles are congruent, we can set the expressions for the two angles equal to each other: $4x + 12 = 5x - 18$

2. Solve for $x$: $4x + 12 = 5x - 18$ Subtract $4x$ from both sides: $12 = x - 18$ Add 18 to both sides: $x = 30$

3. Now substitute $x = 30$ into the expression for $m \angle MPQ$: $m \angle MPQ = 4x + 12 = 4(30) + 12 = 120 + 12 = 132^\circ$

Thus, the measure of $\angle MPQ$ is $132^\circ$.

Would you like more details on the process, or do you have any further questions?

Follow-up Questions:

1. What are vertical angles, and why are they always congruent?
2. Can you explain how to identify supplementary angles in a diagram like this?
3. If another angle was given as supplementary to $\angle MPQ$, how would you solve for it?
4. How does the transversality of lines affect angle relationships in the figure?
5. How would you solve the problem if the given angles were not vertical angles?

Tip: When solving problems with vertical angles, always remember they are congruent, which simplifies solving for unknowns!