To solve for $x$ and $z$ in this figure, we need to analyze the relationships between the angles given the parallel lines and the transversals.

1. Understanding the Angle Relationships: Since $m \parallel n$ and $p \parallel q$, we can apply the properties of corresponding angles and alternate interior angles formed by parallel lines intersected by transversals.

2. Setting Up Equations:

   • The angle labeled $64^\circ$ and the angle $x^\circ$ are corresponding angles because of the parallel lines $m$ and $n$ with transversal $p$. Therefore, $x = 64^\circ$.
   • The angle $4z - 48$ is an alternate interior angle to the $64^\circ$ angle, due to the parallel lines $p$ and $q$ with transversal $n$. This gives us the equation: $4z - 48 = 64$

3. Solving for $z$: Rearranging the equation: $4z = 64 + 48$ $4z = 112$ $z = 28$

4. Summary of Solutions:

   • $x = 64^\circ$
   • $z = 28$

Let me know if you need further clarification on any step, or if you'd like a deeper explanation of the angle relationships.

Here are some related questions to deepen your understanding:

1. What are the properties of corresponding angles formed by parallel lines?
2. How do alternate interior angles behave when lines are parallel?
3. Can you identify other pairs of corresponding angles in the diagram?
4. How would the solution change if one of the lines was not parallel?
5. What other properties of parallel lines are helpful in geometry problems?

Tip: Always look for parallel lines and transversals to identify possible angle relationships when solving geometry problems involving lines and angles.