To solve for the measures of angles $\angle RSQ$ and $\angle TSQ$, we need to make use of the fact that:

1. $\angle RST$ is a right angle, so the sum of $\angle RSQ$ and $\angle TSQ$ should be $90^\circ$.
2. The measures of $\angle RSQ$ and $\angle TSQ$ are given as expressions in terms of $x$:
   • $m\angle RSQ = (15x - 43)^\circ$
   • $m\angle TSQ = (8x + 18)^\circ$

Thus, we can set up the following equation:

$(15x - 43) + (8x + 18) = 90$

Step 1: Simplify the equation

$15x - 43 + 8x + 18 = 90$ $23x - 25 = 90$

Step 2: Solve for $x$

Add 25 to both sides:

$23x = 115$

Now, divide both sides by 23:

$x = 5$

Step 3: Find $m\angle RSQ$ and $m\angle TSQ$

Now that we have $x = 5$, substitute this value into the expressions for $m\angle RSQ$ and $m\angle TSQ$.

$m\angle RSQ = 15(5) - 43 = 75 - 43 = 32^\circ$ $m\angle TSQ = 8(5) + 18 = 40 + 18 = 58^\circ$

Final Answer:

• $m\angle RSQ = 32^\circ$
• $m\angle TSQ = 58^\circ$

Would you like more details or have any questions?

Here are some follow-up questions to consider:

1. How would the equation change if the angle sum was not $90^\circ$?
2. What if the expressions for the angles were quadratic instead of linear?
3. How can we verify that the angles are supplementary to the right angle?
4. What are other methods to solve for the angles without algebraic expressions?
5. How can this type of angle problem apply to real-world scenarios?

Tip: Always double-check if your angles should sum up to a specific value, especially when dealing with geometric figures like triangles or right angles.