The image shows two intersecting lines, forming an "X" shape, and the angles around the intersection are labeled with $z^\circ$ at the top, $x^\circ$ and $y^\circ$ on the sides.

To find the value of $z$ in terms of $x$ and $y$, we can use the property of vertically opposite angles and linear pair angles.

Steps:

1. Vertically Opposite Angles: The angle labeled $z^\circ$ is vertically opposite to another angle (directly across from it). Vertically opposite angles are equal, so both angles at the intersection point marked as $z^\circ$ are the same.

2. Linear Pair: The angles on a straight line sum to 180°. Therefore, on the left and right of the intersection, we can write: $z + y = 180^\circ$ and $z + x = 180^\circ.$

Solving for $z$:

From either equation, we can solve for $z$. Let's use the equation $z + y = 180^\circ$: $z = 180^\circ - y$

Thus, the value of $z$ in terms of $y$ is: $z = 180^\circ - y$

Similarly, from $z + x = 180^\circ$: $z = 180^\circ - x$

Therefore, $z$ depends on whether you relate it to $x$ or $y$.

Would you like any further explanation or details on this? Here are some related questions you could explore:

1. What are vertically opposite angles, and why are they equal?
2. Can you derive the same result by using angle sum properties?
3. How would this change if $x = y$?
4. What other properties of intersecting lines could be useful here?
5. Could we find the measure of $z$ if specific values for $x$ and $y$ were given?

Tip: Always remember that angles formed by two intersecting lines have symmetrical properties like vertically opposite and linear pairs!