We are given a triangle $\triangle VWX$, with $\overline{VX}$ extended through point $X$ to point $Y$. The following angles are provided:

• $m\angle XVW = (x + 6)^\circ$,
• $m\angle WXY = (8x - 9)^\circ$,
• $m\angle VWX = (2x + 10)^\circ$.

Since $\overline{VX}$ is extended to point $Y$, we can use the Exterior Angle Theorem, which states that the exterior angle (in this case $\angle WXY$) is equal to the sum of the two non-adjacent interior angles of the triangle $\triangle VWX$.

Thus, we have the equation: $m\angle WXY = m\angle XVW + m\angle VWX$

Substituting the given expressions: $(8x - 9) = (x + 6) + (2x + 10)$

Now, let's simplify and solve for $x$:

1. Combine like terms on the right-hand side: $8x - 9 = (x + 2x) + (6 + 10)$ $8x - 9 = 3x + 16$

2. Subtract $3x$ from both sides: $8x - 3x - 9 = 16$ $5x - 9 = 16$

3. Add 9 to both sides: $5x = 25$

4. Divide by 5: $x = 5$

Thus, the value of $x$ is $\boxed{5}$.

Would you like further details on this solution?

Here are some related questions to extend your understanding:

1. What is the measure of $\angle XVW$ when $x = 5$?
2. How would the problem change if the angle $\angle WXY$ were different?
3. Can you explain the Exterior Angle Theorem in your own words?
4. What if we were given the sum of all interior angles? How would that affect solving for $x$?
5. How do linear pair properties apply in this kind of problem?

Tip: Always remember that in any triangle, the sum of the interior angles is 180°, which can help check or solve for missing angles.