In ΔSTU,  start overline, S, U, end overline*SU* is extended through point U to point V,  m, angle, U, S, T, equals, left parenthesis, x, minus, 6, right parenthesis, degreesm∠*UST*=(*x*−6)∘,  m, angle, T, U, V, equals, left parenthesis, 4, x, minus, 17, right parenthesis, degreesm∠*TUV*=(4*x*−17)∘,   and m, angle, S, T, U, equals, left parenthesis, x, plus, 7, right parenthesis, degreesm∠*STU*=(*x*+7)∘.   Find m, angle, S, T, U, .m∠*STU*.

To find \( m\angle STU \) (the measure of \( \angle STU \)) in the triangle \( \Delta STU \), we need to first find the value of \( x \). 

Given:
- \( m\angle UST = (x - 6)^\circ \)
- \( m\angle TUV = (4x - 17)^\circ \) (exterior angle)
- \( m\angle STU = (x + 7)^\circ \)

The exterior angle theorem states that the exterior angle is equal to the sum of the two opposite interior angles. Therefore, we have:
\[
m\angle TUV = m\angle UST + m\angle STU
\]

Substituting the given values:
\[
4x - 17 = (x - 6) + (x + 7)
\]

First, simplify the right-hand side:
\[
4x - 17 = x - 6 + x + 7
\]
\[
4x - 17 = 2x + 1
\]

Now, solve for \( x \):
\[
4x - 2x = 1 + 17
\]
\[
2x = 18
\]
\[
x = \frac{18}{2} = 9
\]

Now that we have \( x = 9 \), we can substitute it back to find \( m\angle STU \):
\[
m\angle STU = x + 7 = 9 + 7 = 16^\circ
\]

So, the measure of \( \angle STU \) is \( \boxed{16^\circ} \).

Would you like any more details or have any further questions?

Here are five related questions for practice:
1. What is the measure of \( \angle UST \) when \( x = 9 \)?
2. What is the measure of \( \angle TUV \) when \( x = 9 \)?
3. Verify the solution by checking if the sum of \( m\angle UST \) and \( m\angle STU \) equals \( m\angle TUV \).
4. If \( SU \) was not extended, how would the problem differ?
5. How do you apply the exterior angle theorem in different types of triangles?

**Tip:** Always double-check your final answers by substituting the value back into the original equations to ensure all relationships hold true.

Learn how to find the measure of angle STU in triangle STU using the Exterior Angle Theorem. Detailed step-by-step solution provided with calculations and verification. Suitable for high school students studying geometry.

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