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To solve this problem, we need to find the measure of angle \( m \angle RST \).

### Step 1: Analyze the given information

- \( m \angle CST = 40^\circ \)
- \( m \angle RSC = 25x - 5 \)
- \( m \angle RST = 32x \)

The angles in question are part of a linear pair (along a straight line). Therefore, the sum of the angles along line \( CS \) is equal to \( 180^\circ \).

So:
\[
m \angle CST + m \angle RSC + m \angle RST = 180^\circ
\]

### Step 2: Set up the equation
Substituting the given angle expressions:

\[
40^\circ + (25x - 5) + 32x = 180^\circ
\]

### Step 3: Simplify the equation
Combine like terms:

\[
40^\circ - 5^\circ + 25x + 32x = 180^\circ
\]
\[
35^\circ + 57x = 180^\circ
\]

### Step 4: Solve for \( x \)
Subtract \( 35^\circ \) from both sides:

\[
57x = 145^\circ
\]

Now, solve for \( x \):

\[
x = \frac{145^\circ}{57} \approx 2.54
\]

### Step 5: Find \( m \angle RST \)
Substitute \( x \approx 2.54 \) into \( m \angle RST = 32x \):

\[
m \angle RST = 32 \times 2.54 \approx 81.3^\circ
\]

Thus, \( m \angle RST \approx 81.3^\circ \).

Would you like further details or have any questions?

Here are some related questions:
1. How would you calculate \( m \angle RSC \) using the value of \( x \)?
2. Why do the angles along a straight line sum up to \( 180^\circ \)?
3. What is the significance of linear pairs in geometry?
4. How would the solution change if \( m \angle CST \) were a different value?
5. How can you confirm the accuracy of the calculated \( m \angle RST \)?

**Tip:** Always check if the sum of the angles is consistent with the given geometry, especially when dealing with linear pairs or triangles.

Find m∠RST if m∠CST = 40°, m∠RSC = 25x - 5, and m∠RST = 32x.

Solve for the measure of angle RST using the given angles and algebraic expressions. This geometry problem applies the linear pair theorem, where the sum of angles along a straight line equals 180°. Suitable for students in Grades 8-10.

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