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To solve for \( m \angle VWX \) in this problem, we can use the fact that the sum of the measures of the interior angles in a triangle is \( 180^\circ \), and also make use of the exterior angle property. The exterior angle at point \( X \), \( m \angle WXY \), is equal to the sum of the measures of the two remote interior angles, \( m \angle VWX \) and \( m \angle XVW \).

Let's set up the equation:

1. **Identify the given angles:**
   - \( m \angle VWX = (2x + 4)^\circ \)
   - \( m \angle WXY = (7x - 19)^\circ \)
   - \( m \angle XVW = (2x + 16)^\circ \)

2. **Use the exterior angle property:**  
   According to the exterior angle theorem, the measure of \( m \angle WXY \) (the exterior angle) is equal to the sum of \( m \angle VWX \) and \( m \angle XVW \) (the two non-adjacent interior angles). Therefore:
   \[
   m \angle WXY = m \angle VWX + m \angle XVW
   \]

3. **Substitute the given expressions into the equation:**
   \[
   7x - 19 = (2x + 4) + (2x + 16)
   \]

4. **Solve the equation for \( x \):**
   \[
   7x - 19 = 4x + 20
   \]
   \[
   7x - 4x = 20 + 19
   \]
   \[
   3x = 39
   \]
   \[
   x = 13
   \]

5. **Calculate \( m \angle VWX \):**
   Substitute \( x = 13 \) back into the expression for \( m \angle VWX \):
   \[
   m \angle VWX = 2x + 4 = 2(13) + 4 = 26 + 4 = 30^\circ
   \]

Therefore, \( m \angle VWX = 30^\circ \).

Would you like further details on any of the steps, or have additional questions?

Here are some related questions for further exploration:
1. What would be the value of \( m \angle XVW \) if \( x = 13 \)?
2. How does the exterior angle theorem help in solving for unknown angle measures?
3. What is the sum of the interior angles in a triangle, and why?
4. Can the exterior angle be smaller than both remote interior angles? Why or why not?
5. How would you solve for \( x \) if a different angle relationship were given?

**Tip:** Remember that the exterior angle of a triangle is always equal to the sum of the two non-adjacent interior angles. This property can simplify finding unknown angles in many triangle problems.

In ΔVWX, VX is extended through point X to point Y, m∠VWX = (2x + 4)°, m∠WXY = (7x - 19)°, and m∠XVW = (2x + 16)°. Find m∠VWX.

In this problem, you will learn how to use the exterior angle theorem to find the measure of m∠VWX in ΔVWX where m∠VWX = (2x + 4)°, m∠WXY = (7x - 19)°, and m∠XVW = (2x + 16)°. Step-by-step explanation included for Grades 9-10.

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