To solve for $x$ and $z$ in this problem, we need to analyze the given figure, which shows two intersecting lines forming vertical angles. Vertical angles are congruent, meaning the angles opposite each other have equal measures.

Step 1: Set Up Equations Using Vertical Angles

The figure shows:

1. An angle labeled as $(7x - 18)^\circ$.
2. A vertically opposite angle labeled as $(8x - 34)^\circ$.
3. An angle labeled $z^\circ$, which is also vertically opposite to another angle that isn’t directly given.

Since vertical angles are equal, we can set up an equation with the expressions for the angles: $7x - 18 = 8x - 34$

Step 2: Solve for $x$

Rearrange the equation to isolate $x$: $7x - 8x = -34 + 18$ $-x = -16$ $x = 16$

Step 3: Find $z$

Once $x = 16$ is found, we can substitute it back into one of the angle expressions to determine the measure of that angle. Let's use $7x - 18$: $7(16) - 18 = 112 - 18 = 94^\circ$ Since $z$ is vertically opposite to this angle, $z = 94^\circ$ as well.

Final Answer

• $x = 16$
• $z = 94^\circ$

Would you like further details on each step or have any questions?

Here are some related questions to consider:

1. What are vertical angles, and why are they congruent?
2. How would the solution change if the given angles were not vertical?
3. Can these types of problems involve supplementary angles instead?
4. How do we confirm the solution is correct using substitution?
5. What is the significance of intersecting lines in angle calculations?

Tip: Always double-check your answers by substituting values back into the original expressions to verify the consistency of the solution.