In this problem, we are dealing with an extended triangle $\triangle TUV$, and we need to solve for $x$ based on the provided angle measures.

Given:

1. $m\angle TUV = (3x + 11)^\circ$
2. $m\angle VTU = (x + 13)^\circ$
3. $m\angle UVW = (7x - 12)^\circ$

Explanation:

Since $TV$ is extended through point $V$ to point $W$, we have an exterior angle $\angle UVW$. The exterior angle theorem states that the measure of the exterior angle is equal to the sum of the two non-adjacent interior angles.

So, we can write the equation:

$m\angle UVW = m\angle TUV + m\angle VTU$

Substituting the given values:

$(7x - 12) = (3x + 11) + (x + 13)$

Solve for $x$:

1. Combine like terms on the right-hand side: $(7x - 12) = 3x + 11 + x + 13$ $(7x - 12) = 4x + 24$

2. Subtract $4x$ from both sides: $3x - 12 = 24$

3. Add 12 to both sides: $3x = 36$

4. Divide by 3: $x = 12$

Thus, the value of $x$ is $\boxed{12}$.

Would you like more details on how this theorem works or have any questions? Here are some additional problems to expand your understanding:

1. What is the sum of the interior angles of any triangle?
2. Can you apply the exterior angle theorem to any polygon, or is it specific to triangles?
3. If the value of $x$ were negative, how would the solution change?
4. What would happen if the triangle were isosceles? How would it affect the angle measures?
5. How can you verify that the exterior angle theorem applies in other geometric scenarios?

Tip: Always remember that the exterior angle theorem is a powerful tool for solving triangle problems where extensions are involved.