To find the measure of $\angle PRQ$, we need to use the information given in the diagram. Here is the step-by-step solution:

1. Identify the angles:

   • $\angle PRQ$ is the angle we need to find.
   • $\angle PQR = \alpha$
   • $\angle QRP = 3\alpha + 6^\circ$
   • $\angle RPS = 48^\circ$

2. Use the fact that the sum of angles in a triangle is $180^\circ$: In $\triangle PQR$: $\alpha + (3\alpha + 6^\circ) + 48^\circ = 180^\circ$

3. Combine like terms and solve for $\alpha$: $\alpha + 3\alpha + 6^\circ + 48^\circ = 180^\circ$ $4\alpha + 54^\circ = 180^\circ$ $4\alpha = 180^\circ - 54^\circ$ $4\alpha = 126^\circ$ $\alpha = \frac{126^\circ}{4}$ $\alpha = 31.5^\circ$

4. Now, find $\angle PRQ$: $\angle PRQ = \alpha = 31.5^\circ$

So, the measure of $\angle PRQ$ is $31.5^\circ$.

Would you like further details or have any questions?

Here are some related questions you might be interested in:

1. How do you solve for an unknown angle in a triangle when given two angles?
2. What are the properties of angles in a triangle?
3. How do you determine the type of triangle based on its angles?
4. What is the sum of the interior angles of a triangle?
5. How can you use the exterior angle theorem to solve for unknown angles?
6. What are complementary and supplementary angles?
7. How do you apply the Pythagorean theorem in solving triangle problems?
8. What are the different types of triangles based on angles and sides?

Tip: Always remember that the sum of the interior angles of a triangle is $180^\circ$, which is a fundamental property useful in solving many geometry problems.