https://file.aiquickdraw.com/mathbot/file/17300952591382n317pbn.jpg

Let's analyze and solve the problem step-by-step.

In this problem, we have:
1. **Given Angles:**
   - \( m \angle UST = (2x - 6)^\circ \)
   - \( m \angle STU = (2x + 18)^\circ \)
   - \( m \angle TUV = (6x - 8)^\circ \)

2. **Angle Relationships:**
   Since \( SU \) is extended through point \( U \) to point \( V \), \( \angle TUV \) is an exterior angle to \( \triangle STU \). According to the exterior angle theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.

   Therefore:
   \[
   m \angle TUV = m \angle UST + m \angle STU
   \]

3. **Setting up the Equation:**
   Substitute the given expressions for each angle:
   \[
   (6x - 8) = (2x - 6) + (2x + 18)
   \]

4. **Solving the Equation:**
   Simplify the right side:
   \[
   (6x - 8) = 4x + 12
   \]

   Now, isolate \( x \):
   \[
   6x - 4x = 12 + 8
   \]
   \[
   2x = 20
   \]
   \[
   x = 10
   \]

### Answer:
The value of \( x \) is \( \boxed{10} \).

Would you like more details, or do you have any questions about this solution?

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Here are five related questions that can expand your understanding:

1. What is the measure of \( \angle UST \) when \( x = 10 \)?
2. What is the measure of \( \angle STU \) when \( x = 10 \)?
3. What is the measure of \( \angle TUV \) when \( x = 10 \)?
4. Can you explain the exterior angle theorem in your own words?
5. How would the solution change if \( \angle TUV \) had a different expression?

**Tip:** Remember, in geometry problems involving triangles, the exterior angle theorem is a powerful tool to solve for unknown values by connecting exterior and interior angles.

In ΔSTU, SU is extended through point U to point V, m∠UST = (2x - 6)°, m∠STU = (2x + 18)°, and m∠TUV = (6x - 8)°. What is the value of x?

Solve for x in triangle ΔSTU where SU is extended to point V, using the Exterior Angle Theorem. The problem involves angle measurements: m∠UST = (2x - 6)°, m∠STU = (2x + 18)°, and m∠TUV = (6x - 8)°. Step-by-step solution provided.

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