To solve for the angles $x$, $y$, and $z$ in the parallelogram $ABCD$, let’s analyze the figure step-by-step.

1. Since $ABCD$ is a parallelogram, opposite angles are equal. Therefore:

   • $\angle DAB = \angle BCD$
   • $\angle ABC = \angle CDA$

2. From the diagram, we know:

   • $\angle ABC = 70^\circ$
   • $\angle BCD = 38^\circ$

3. Since $\angle BCD = 38^\circ$, by the property of a parallelogram, the opposite angle $\angle DAB$ (or angle $z$) must also be $38^\circ$.

   Therefore: $z = 38^\circ$

4. For a parallelogram, adjacent angles are supplementary. This means: $\angle ABC + \angle CDA = 180^\circ$ Given that $\angle ABC = 70^\circ$, we can find $\angle CDA$ (angle $x$) as follows: $x = 180^\circ - 70^\circ = 110^\circ$

5. Since $\angle DAB$ and $\angle BCD$ are also opposite angles in the parallelogram, $y = 70^\circ$.

To summarize:

• $x = 110^\circ$
• $y = 70^\circ$
• $z = 38^\circ$

Would you like a more detailed explanation or have any further questions?

Here are 5 related questions to explore further:

1. What are the properties of a parallelogram regarding its angles and sides?
2. How would you find angles in a different type of quadrilateral, like a trapezoid?
3. How would this problem change if $ABCD$ was a rectangle?
4. What role do supplementary angles play in solving problems involving parallelograms?
5. How would you use similar methods to find unknown angles in triangles?

Tip: In parallelograms, remember that opposite angles are always equal, and adjacent angles are supplementary. This can help in solving for unknown angles quickly.