To solve this problem, we need to find the measure of $\angle RPQ$ given that the measure of $\angle RPQ = [5x + 13]^\circ$.

The figure shows that two angles, $\angle RPM = 4x - 3$ and $\angle MPQ = 2x + 2$, are adjacent angles that together form $\angle RPQ$.

Step-by-Step Solution:

1. Understanding the Problem:

   • $\angle RPQ = \angle RPM + \angle MPQ$
   • $\angle RPM = 4x - 3$
   • $\angle MPQ = 2x + 2$
   • $\angle RPQ = 5x + 13$

2. Setting Up the Equation: Since $\angle RPQ$ is the sum of $\angle RPM$ and $\angle MPQ$, we can write:

   $(4x - 3) + (2x + 2) = 5x + 13$

3. Simplifying the Equation:

   $4x - 3 + 2x + 2 = 5x + 13$

   Combine like terms:

   $6x - 1 = 5x + 13$

4. Solving for $x$:

   Subtract $5x$ from both sides:

   $6x - 5x - 1 = 13$

   $x - 1 = 13$

   Add 1 to both sides:

   $x = 14$

5. Finding $m\angle RPQ$:

   Substitute $x = 14$ back into the expression for $\angle RPQ$:

   $m\angle RPQ = 5x + 13 = 5(14) + 13$

   Calculate the value:

   $m\angle RPQ = 70 + 13 = 83$

Therefore, the measure of $\angle RPQ$ is $\boxed{83}$ degrees.

Answer:

The correct answer is 83.

Would you like more details on how this solution was derived or have any questions?

Additional Related Questions:

1. How would the calculation change if $\angle RPQ = [5x + 15]^\circ$?
2. What is the effect on the angles if the value of $x$ were to be a negative number?
3. How can you use the properties of supplementary and complementary angles in similar problems?
4. What is the importance of combining like terms when solving linear equations?
5. How can you check your solution to ensure it's correct?

Tip: Always double-check each step in algebraic manipulation to avoid small errors that can affect the final answer.