We are given two pairs of vertically opposite angles in the figure. Based on the properties of vertically opposite angles, they are always equal.

From the figure:

• $z^\circ$ is vertically opposite to $(9x - 26)^\circ$, so: $z = 9x - 26$

• $(11x - 74)^\circ$ is vertically opposite to $(9x - 26)^\circ$, so: $11x - 74 = 9x - 26$

Step 1: Solve for $x$

We have the equation: $11x - 74 = 9x - 26$

First, subtract $9x$ from both sides: $2x - 74 = -26$

Next, add 74 to both sides: $2x = 48$

Now, divide by 2: $x = 24$

Step 2: Solve for $z$

Substitute $x = 24$ into the equation for $z$: $z = 9(24) - 26$ $z = 216 - 26$ $z = 190$

Final Answers:

• $x = 24$
• $z = 190^\circ$

Let me know if you would like further explanation on any step or additional details!

Here are 5 related questions you can try:

1. If $z = 190^\circ$, what is the value of the other angle opposite to it?
2. How do you identify vertically opposite angles in other intersecting line figures?
3. What is the sum of all angles at the intersection of two straight lines?
4. How would the result change if the angles were complementary instead of vertically opposite?
5. Can you find the missing angles if given only one angle at the intersection?

Tip: In geometry, always remember that vertically opposite angles are congruent, which can help you solve for unknown values easily!