https://file.aiquickdraw.com/mathbot/file/1726155269935zm1f6kuq.jpg

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.

Given m∠QPS = 40°, m∠RPS = 8x + 7°, and m∠QPR = 9x + 16°. Find m∠QPR.

Learn how to find the measure of angle QPR given m∠QPS = 40°, m∠RPS = 8x + 7°, and m∠QPR = 9x + 16°. This problem covers geometric concepts and algebraic solutions, ideal for students in Grades 8-10.

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