In the given diagram, we have the following angles:

1. $m \angle QPS = 40^\circ$
2. $m \angle RPS = 8x + 7^\circ$
3. $m \angle QPR = 9x + 16^\circ$

The objective is to find $m \angle QPR$.

Solution:

Since the points $Q$, $P$, $R$, and $S$ seem to form a linear set of angles at point $P$, we can assume that: $m \angle QPS + m \angle RPS = 180^\circ$ This relationship exists because $\angle QPS$ and $\angle RPS$ are adjacent angles forming a straight line.

Step 1: Set up the equation

$m \angle QPS + m \angle RPS = 180^\circ$ $40^\circ + (8x + 7^\circ) = 180^\circ$

Step 2: Simplify the equation

$40^\circ + 8x + 7^\circ = 180^\circ$ $47^\circ + 8x = 180^\circ$ $8x = 180^\circ - 47^\circ$ $8x = 133^\circ$ $x = \frac{133^\circ}{8}$ $x = 16.625$

Step 3: Find $m \angle QPR$

Now substitute the value of $x = 16.625$ into the expression for $m \angle QPR$: $m \angle QPR = 9x + 16^\circ$ $m \angle QPR = 9(16.625) + 16^\circ$ $m \angle QPR = 149.625 + 16^\circ$ $m \angle QPR = 165.625^\circ$

Therefore, the measure of $m \angle QPR$ is approximately $165.6^\circ$.

Would you like further details or clarifications on this solution?

Here are 5 additional questions you might consider:

1. How do adjacent angles on a straight line sum to 180°?
2. What happens if two angles are complementary instead of supplementary?
3. Can you solve for $x$ in different angle relationships like vertical angles?
4. How would you find angles in a triangle using similar algebraic methods?
5. What is the impact of transversals on angle calculations in parallel lines?

Tip: When working with angles, always check if they form a straight line or other geometric relationships to set up correct equations.