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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.

In ΔRST, m∠R = (2x - 12)°, m∠S = (8x - 5)°, and m∠T = (x - 12)°. Find m∠S.

Learn how to find m∠S in triangle ΔRST where the angles are given as algebraic expressions in terms of x. Using the Triangle Sum Theorem, this step-by-step solution involves algebraic simplification and solving for x. Suitable for students in Grades 8-10.

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