We are given that $w$ lies in the interior of angle $\angle XYZ$, meaning that:

$\angle XYZ = \angle XYW + \angle WYZ$

We know:

$\angle XYZ = 139^\circ$ $\angle XYW = 7x - 2$ $\angle WYZ = 2x + 15$

Now, using the fact that $\angle XYZ = \angle XYW + \angle WYZ$, we can set up the equation:

$139 = (7x - 2) + (2x + 15)$

Simplifying the equation:

$139 = 7x - 2 + 2x + 15$ $139 = 9x + 13$

Now, subtract 13 from both sides:

$139 - 13 = 9x$ $126 = 9x$

Now, divide both sides by 9:

$x = \frac{126}{9}$ $x = 14$

Thus, $x = 14$.

Would you like further clarification or details?

Here are 5 related questions to expand on this concept:

1. How would you find the value of $\angle XYW$ and $\angle WYZ$ using $x = 14$?
2. What happens if point $w$ lies on the angle bisector of $\angle XYZ$?
3. Can you use this method to solve for angles in non-linear figures?
4. How do complementary and supplementary angles relate to problems like this?
5. What would the solution be if $\angle XYZ$ was a right angle?

Tip: Always double-check your equation setup, ensuring all angle relationships are properly accounted for.