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The problem presented involves triangle \( \Delta VWX \), where line segment \( \overline{VX} \) is extended through point \( X \) to point \( Y \). The angles are given as follows:

- \( m\angle VWX = (2x + 4)^\circ \),
- \( m\angle WXY = (7x - 19)^\circ \),
- \( m\angle XVW = (2x + 16)^\circ \).

We are asked to find \( m\angle VWX \).

### Approach:

This is a case where the **Exterior Angle Theorem** can be applied. According to this theorem, the exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.

Here, \( \angle WXY \) is the exterior angle at point \( X \) for triangle \( \Delta VWX \), and it should be equal to the sum of the interior angles \( \angle VWX \) and \( \angle XVW \).

Thus, using the Exterior Angle Theorem:

\[
m\angle WXY = m\angle VWX + m\angle XVW
\]

Substitute the given values:

\[
(7x - 19) = (2x + 4) + (2x + 16)
\]

Simplify the right-hand side:

\[
(7x - 19) = 2x + 4 + 2x + 16
\]
\[
(7x - 19) = 4x + 20
\]

Now, solve for \( x \):

\[
7x - 4x = 20 + 19
\]
\[
3x = 39
\]
\[
x = 13
\]

### Finding \( m\angle VWX \):

Now substitute \( x = 13 \) into the expression for \( m\angle VWX \):

\[
m\angle VWX = 2x + 4 = 2(13) + 4 = 26 + 4 = 30^\circ
\]

### Final Answer:
\[
m\angle VWX = 30^\circ
\]

Let me know if you'd like further clarification or have any questions. Here are some related questions to explore further:

1. What is the measure of \( m\angle XVW \) given \( x = 13 \)?
2. Can you find the value of \( m\angle WXY \) with \( x = 13 \)?
3. What is the significance of the Exterior Angle Theorem in different triangle types?
4. How would the solution change if one of the given angle expressions was incorrect?
5. Can you derive the sum of the interior angles of any triangle using the exterior angle theorem?

**Tip:** Always check if a problem involves exterior angles, as they provide a straightforward way to relate angles in geometric figures.

In triangle VWX, VX is extended through point X to point Y. The measures of the angles are given as m∠VWX = (2x + 4)°, m∠WXY = (7x - 19)°, and m∠XVW = (2x + 16)°. Find m∠VWX.

Learn how to find the angle m∠VWX in triangle VWX by applying the Exterior Angle Theorem. This problem involves algebraic expressions for angles and is suitable for students in Grades 9-12. The key concepts include geometry and algebra.

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