https://file.aiquickdraw.com/mathbot/file/17286199476856f0c9ia3.jpg

The image you uploaded appears to show a triangle \( \triangle TSU \) with an exterior angle at point \( S \), with the following information:

- \( \angle TSU = 58^\circ \)
- \( \angle T = 5x + 10 \) (expression for angle \( T \))
- \( \angle U = 11x + 2 \) (expression for angle \( U \))

### Steps to Solve for \( x \):

Using the exterior angle theorem, we know that the exterior angle \( \angle TSU = \angle T + \angle U \). Therefore, the equation becomes:

\[
58^\circ = (5x + 10) + (11x + 2)
\]

Simplifying the equation:

\[
58 = 5x + 10 + 11x + 2
\]
\[
58 = 16x + 12
\]

Now, subtract 12 from both sides:

\[
58 - 12 = 16x
\]
\[
46 = 16x
\]

Now, divide both sides by 16:

\[
x = \frac{46}{16} = 2.875
\]

So, \( x = 2.875 \).

### Verifying Angles:
Now, let's plug \( x = 2.875 \) back into the expressions for \( \angle T \) and \( \angle U \) to verify:

\[
\angle T = 5(2.875) + 10 = 14.375 + 10 = 24.375^\circ
\]
\[
\angle U = 11(2.875) + 2 = 31.625 + 2 = 33.625^\circ
\]

Summing the two interior angles:

\[
24.375^\circ + 33.625^\circ = 58^\circ
\]

This confirms that the solution is correct.

Would you like further details on any part of the solution, or help with other problems?

### Follow-up Questions:
1. What is the relationship between interior and exterior angles of a triangle?
2. How does the exterior angle theorem help in solving for unknown angles?
3. Can you find the sum of interior angles for any triangle? How?
4. How would the problem change if we were given the sides instead of the angles?
5. What are some real-world applications of the exterior angle theorem?

**Tip:** Always remember that the sum of interior angles in any triangle is \( 180^\circ \), which can help verify your calculations!

Solve for x in the triangle TSU where angle T = 5x + 10, angle U = 11x + 2, and exterior angle TSU = 58°.

Learn how to solve for x in a triangle using the exterior angle theorem. This problem covers geometry and algebra concepts, where angle T = 5x + 10, angle U = 11x + 2, and the exterior angle TSU = 58°. Suitable for students in Grades 8-10.

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