The image shows an angle relationship problem involving intersecting lines. We are asked to find the measure of angle $\angle WXZ$.

Key details:

• $\angle WXY = 5y^\circ$
• $\angle YXZ = (2y + 78)^\circ$

Since these two angles are vertically opposite (as formed by intersecting lines), their measures must be equal. Therefore, we can set up the equation:

$5y = 2y + 78$

Now, let's solve for $y$:

$5y - 2y = 78$

$3y = 78$

$y = \frac{78}{3} = 26$

Next, we can substitute $y = 26$ back into either expression to find the measure of $\angle WXZ$. Let's substitute it into $5y$:

$\angle WXZ = 5(26) = 130^\circ$

Thus, the measure of $\angle WXZ$ is $130^\circ$.

Would you like further details or have any other questions?

Here are 5 related questions to expand your understanding:

1. How do we know that vertically opposite angles are equal?
2. What other pairs of angles could be equal in this diagram?
3. How would the solution change if $\angle YXZ$ was expressed differently?
4. Can you identify other angle relationships (like supplementary angles) from the diagram?
5. If the lines were not straight, how would this affect the angle relationships?

Tip: Always check for symmetrical properties like vertically opposite or alternate interior angles when dealing with intersecting lines.