The problem asks to find the measures of angles $\angle RSQ$ and $\angle TSQ$, given the following information:

• $\angle RSQ = (15x - 43)^\circ$
• $\angle TSQ = (8x + 18)^\circ$
• The figure shows a right angle at $S$, meaning $\angle RST = 90^\circ$.

Since these three angles are part of the same straight line at $S$, we know that:

$\angle RSQ + \angle TSQ + \angle RST = 180^\circ$

Substituting the known values:

$(15x - 43)^\circ + (8x + 18)^\circ + 90^\circ = 180^\circ$

Now, let's solve for $x$:

1. Combine like terms:

$15x - 43 + 8x + 18 + 90 = 180$

$23x + 65 = 180$

2. Subtract 65 from both sides:

$23x = 115$

3. Divide by 23:

$x = 5$

Now that we have $x = 5$, we can find the measure of each angle:

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

Thus:

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

Would you like further clarification or details?

Related Questions:

1. What is the sum of interior angles in a triangle, and how is it used in geometric proofs?
2. How would the answer change if the angles formed a different type of triangle?
3. What are complementary and supplementary angles, and how are they applied here?
4. How would you find the angles if the sum was greater than 180° due to a different type of figure?
5. What happens if one of the given expressions involves a quadratic equation?

Tip:

When dealing with angles on a straight line, always remember that the sum of the angles will equal 180°. This is useful in many geometric problems!