To solve for the values of $x$ and $z$ in the given diagram, we can use the concept of vertical angles. Vertical angles are always equal.

Step 1: Set the equations equal

The angles $(12x + 5)^\circ$ and $(14x - 13)^\circ$ are vertical angles, so they must be equal:

$12x + 5 = 14x - 13$

Step 2: Solve for $x$

To solve for $x$, we will first move all the terms involving $x$ to one side and the constant terms to the other side:

$12x - 14x = -13 - 5$

$-2x = -18$

Now, divide both sides by $-2$:

$x = 9$

Step 3: Find $z$

The angle $z^\circ$ is also vertical to one of the angles in the diagram, so it will be equal to either $(12x + 5)^\circ$ or $(14x - 13)^\circ$.

Let's substitute $x = 9$ into the equation $12x + 5$:

$12(9) + 5 = 108 + 5 = 113^\circ$

Therefore, $z = 113^\circ$.

Final Answer:

• $x = 9$
• $z = 113^\circ$

Would you like any more details or have any other questions?

Here are five related questions for further exploration:

1. What is the measure of the other pair of vertical angles in the diagram?
2. How would the solution change if the given angles were supplementary instead of vertical?
3. What properties of vertical angles make them equal?
4. Can you apply the concept of vertical angles to a real-world problem?
5. How can we verify the solution for $x$ and $z$ by plugging the values back into the original equations?

Tip: Always check if angles are vertical or supplementary when dealing with intersecting lines, as this can help simplify your problem-solving approach.