In ΔTUV,  start overline, T, V, end overline
TV
is extended through point V to point W,  m, angle, T, U, V, equals, left parenthesis, 3, x, plus, 11, right parenthesis, degreesm∠TUV=(3x+11)
∘
,  m, angle, V, T, U, equals, left parenthesis, x, plus, 13, right parenthesis, degreesm∠VTU=(x+13)
∘
,   and m, angle, U, V, W, equals, left parenthesis, 7, x, minus, 12, right parenthesis, degreesm∠UVW=(7x−12)
∘
.   What is the value of x, question markx?

To solve for \( x \) in the given triangle \( \Delta TUV \), we can use the fact that the exterior angle \( \angle UVW \) is equal to the sum of the two opposite interior angles \( \angle TUV \) and \( \angle VTU \).

Given:
- \( m\angle TUV = (3x + 11)^\circ \)
- \( m\angle VTU = (x + 13)^\circ \)
- \( m\angle UVW = (7x - 12)^\circ \)

By the exterior angle theorem:
\[
m\angle UVW = m\angle TUV + m\angle VTU
\]

Substituting the given values:
\[
7x - 12 = (3x + 11) + (x + 13)
\]

First, simplify the right-hand side:
\[
7x - 12 = 3x + 11 + x + 13
\]
\[
7x - 12 = 4x + 24
\]

Now, solve for \( x \):
\[
7x - 4x = 24 + 12
\]
\[
3x = 36
\]
\[
x = \frac{36}{3} = 12
\]

So, the value of \( x \) is \( \boxed{12} \).

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

Here are five related questions for practice:
1. How do you verify the solution in the context of the triangle?
2. What is the measure of \( \angle TUV \) when \( x = 12 \)?
3. What is the measure of \( \angle VTU \) when \( x = 12 \)?
4. Can the sum of the interior angles of \( \Delta TUV \) provide additional insights?
5. How would the problem change if the triangle were isosceles?

**Tip:** When dealing with angles in triangles, remember that the exterior angle theorem can be a powerful tool for finding unknown angles or variables.

Learn how to solve for x in triangle TUV using the Exterior Angle Theorem. Given angles m∠TUV = (3x + 11)°, m∠VTU = (x + 13)°, and m∠UVW = (7x - 12)°, find the value of x step-by-step. Suitable for high school students studying geometry and triangles.

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