In the given diagram, the lines $k \parallel l$ and $m \parallel n$, indicate a pair of parallel lines cut by two transversals. Let's proceed step-by-step to find the values of $x$ and $z$:

Step 1: Identify corresponding and alternate angles

1. The angle labeled $(6z - 108)^\circ$ and the angle labeled $72^\circ$ form a pair of corresponding angles since the transversal line cuts through parallel lines. Therefore, they are equal.

$6z - 108 = 72$

Step 2: Solve for $z$

Now, solve the equation:

$6z - 108 = 72$ $6z = 72 + 108$ $6z = 180$ $z = \frac{180}{6} = 30$

Step 3: Determine $x$

Since the angle $x^\circ$ forms an alternate interior angle with the angle labeled $72^\circ$, these angles are equal. Therefore:

$x = 72^\circ$

Final Answer:

$x = 72^\circ$ $z = 30$

Would you like a detailed explanation of alternate and corresponding angles or any other clarifications?

Here are 5 related questions you could explore:

1. How do you prove that corresponding angles are congruent in parallel lines?
2. What are alternate interior angles, and why are they equal in parallel lines?
3. How can we solve for unknown angles when transversals intersect parallel lines?
4. What is the significance of parallel lines in geometry?
5. How do the angle sum properties help in solving complex angle problems?

Tip: Always label all known angles in geometric figures; this simplifies the process of finding unknown angles using parallel line properties.