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, plus, 5, right parenthesis, degreesm∠UST=(x+5) 
∘
 ,  m, angle, S, T, U, equals, left parenthesis, 2, x, plus, 16, right parenthesis, degreesm∠STU=(2x+16) 
∘
 ,   and m, angle, T, U, V, equals, left parenthesis, 6, x, minus, 9, right parenthesis, degreesm∠TUV=(6x−9) 
∘
 .   Find m, angle, S, T, U, .m∠STU.

We are given that \( \overline{SU} \) is extended to point \( V \), and we have three angles:

- \( m \angle UST = (x + 5)^\circ \)
- \( m \angle STU = (2x + 16)^\circ \)
- \( m \angle TUV = (6x - 9)^\circ \)

Since \( \overline{SU} \) is extended to point \( V \), \( \angle TUV \) is an **exterior angle** of \( \triangle STU \). By the exterior angle theorem, the measure of an exterior angle is equal to the sum of the two non-adjacent interior angles. Therefore, we have:

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

Substituting the given expressions:

\[
6x - 9 = (x + 5) + (2x + 16)
\]

Simplifying the right-hand side:

\[
6x - 9 = 3x + 21
\]

Now, solve for \( x \):

\[
6x - 3x = 21 + 9
\]
\[
3x = 30
\]
\[
x = 10
\]

Now that we have \( x = 10 \), we can find \( m \angle STU \):

\[
m \angle STU = 2x + 16 = 2(10) + 16 = 20 + 16 = 36^\circ
\]

Thus, \( m \angle STU = 36^\circ \).

Would you like further details or have any other questions?

Here are 5 related questions to explore:

1. What is the value of \( m \angle UST \)?
2. What is the measure of \( m \angle TUV \) with \( x = 10 \)?
3. How do you use the exterior angle theorem in geometry?
4. Can you calculate the sum of interior angles of \( \triangle STU \)?
5. What is the significance of extending a side of a triangle in angle calculations?

**Tip**: The exterior angle theorem is a useful tool for finding unknown angles in triangles, especially when a side is extended.

In ΔSTU, SU is extended through point U to point V. m∠UST = (x + 5)°, m∠STU = (2x + 16)°, and m∠TUV = (6x - 9)°. Find m∠STU.

Learn how to find m∠STU in ΔSTU using the exterior angle theorem. The problem involves algebraic expressions and geometric properties of triangles, suitable for students in Grades 9-10. Step-by-step solution included.

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