Let's analyze the image and solve the problem step by step.

Given:

• The diagram shows angles $\angle SQR = 16^\circ$ and $\angle PQS = (5x - 37)^\circ$.
• $\angle PQS$ and $\angle SQR$ form a straight line, so their sum should be $180^\circ$.

Step 1: Set up the equation

Since $\angle PQS + \angle SQR = 180^\circ$, we can write: $(5x - 37)^\circ + 16^\circ = 180^\circ$

Step 2: Simplify the equation

Combine like terms: $5x - 37 + 16 = 180$ $5x - 21 = 180$

Step 3: Solve for $x$

Add 21 to both sides of the equation: $5x = 201$ Now, divide by 5: $x = 40.2$

Step 4: Verify the equation given

From the multiple choices shown:

1. $9x - 36 = 90$
2. $9x - 40 = 90$
3. $9x + 49 = 90$
4. $9x + 36 = 90$

The equation we derived does not exactly match any of these provided choices, so it seems there might be a mix-up. However, based on the logical derivation of $x = 40.2$ using the correct relationship of the angles, our calculated value should be used.

Final Answer:

The correct value for $x$ is approximately 40.2.

Do you want further details or have any questions? Here are some related questions you might find useful:

1. How do you determine if angles form a straight line?
2. What are the steps to combine like terms in an equation?
3. Why do we add 21 to both sides of the equation in this problem?
4. What are the steps to isolate $x$ in an equation?
5. How do you verify that your solution for $x$ is correct?

Tip: Always double-check the equation you are working with to ensure it matches the scenario described in the problem.