We are given a triangle $\Delta RST$, with the following angles:

$m\angle R = (2x - 12)^\circ, \quad m\angle S = (8x - 5)^\circ, \quad m\angle T = (x - 12)^\circ$

The sum of the angles in any triangle is always $180^\circ$. Therefore, we can set up the following equation based on the sum of the angles:

$(2x - 12) + (8x - 5) + (x - 12) = 180$

Now, let's solve for $x$.

Step 1: Simplify the equation

$2x - 12 + 8x - 5 + x - 12 = 180$ Combine like terms: $(2x + 8x + x) + (-12 - 5 - 12) = 180$ $11x - 29 = 180$

Step 2: Solve for $x$

Add 29 to both sides: $11x = 180 + 29$ $11x = 209$ Now, divide by 11: $x = \frac{209}{11} \approx 19$

Step 3: Find $m\angle S$

Substitute $x = 19$ into the expression for $m\angle S = (8x - 5)^\circ$:

$m\angle S = 8(19) - 5 = 152 - 5 = 147^\circ$

Thus, $m\angle S = 147^\circ$.

Let me know if you'd like further details or have any questions! Here are five related questions to expand your understanding:

1. How can we verify the solution using the values of the other angles?
2. What happens if one angle in the triangle is a right angle?
3. How would the solution change if we were solving for $m\angle R$ instead?
4. Can the angles of a triangle ever be negative? Why or why not?
5. How does the triangle inequality theorem relate to the angles of a triangle?

Tip: Always ensure that the sum of all angles in any triangle equals $180^\circ$, which is a fundamental property of triangles in Euclidean geometry.