We are given that Ray WY is the angle bisector of $\angle XWZ$. This means that the two angles on either side of the bisector, $\angle XWY$ and $\angle ZWY$, are equal.

The two angle expressions are:

$m \angle XWY = 5x - 4$ $m \angle ZWY = 8x - 28$

Since Ray WY bisects $\angle XWZ$, these two angles must be equal:

$5x - 4 = 8x - 28$

Let’s solve for $x$:

1. Subtract $5x$ from both sides: $-4 = 3x - 28$

2. Add 28 to both sides: $24 = 3x$

3. Divide both sides by 3: $x = 8$

Now substitute $x = 8$ back into either angle expression to find $m \angle XWY$:

$m \angle XWY = 5(8) - 4 = 40 - 4 = 36^\circ$

Thus, $m \angle XWY = 36^\circ$.

Would you like a detailed explanation or have further questions?

Here are some related questions to deepen your understanding:

1. What is the measure of $m \angle ZWY$ after solving for $x$?
2. How do we verify the two angles are equal using algebra?
3. How does the concept of angle bisectors apply to other geometric figures?
4. Can angle bisectors divide triangles into two congruent triangles? Why or why not?
5. What properties of angles can we infer when working with polygons?

Tip: When solving equations involving bisectors, always ensure the angles on both sides of the bisector are set equal for proper problem setup.