To solve this problem, let’s analyze the image and the information provided:

Information from the diagram:

1. $\angle PQR = 90^\circ$, since it is marked as a right angle.
2. $\angle PQS = 3x$ (labeled in the diagram).
3. $\angle SQR = (8x + 2)^\circ$ (labeled in the diagram).

Using the fact that the sum of the angles around point $Q$ equals $90^\circ$ (because $\angle PQR$ is a right angle), we can write:

$\angle PQS + \angle SQR = 90^\circ$

Substitute the given expressions:

$3x + (8x + 2) = 90$

Solve for $x$:

$3x + 8x + 2 = 90$ $11x + 2 = 90$ $11x = 88$ $x = 8$

Find $m\angle PQS$:

Substitute $x = 8$ into $\angle PQS = 3x$: $\angle PQS = 3(8) = 24^\circ$

Thus, the measure of $m\angle PQS$ is: $\boxed{24^\circ}$

Would you like a detailed explanation of this solution or have additional questions? Here are some related problems to explore:

1. What is the value of $m\angle SQR$ in this diagram?
2. Can we verify the solution by recalculating the total angle sum?
3. What happens if $\angle PQR$ were not $90^\circ$?
4. Explore the relationship between supplementary and complementary angles in other triangles.
5. How can we solve for unknown angles in polygons using angle relationships?

Tip: Always double-check your work by substituting values back into the original equations!