To solve for $x$, $y$, and $z$ in the parallelogram:

Properties of a Parallelogram:

1. Opposite angles are equal.
2. Adjacent angles are supplementary (add up to $180^\circ$).

From the given diagram:

• $x^\circ$ is opposite $z^\circ$, so $x = z$.
• $y^\circ$ is adjacent to $x^\circ$, and their sum is $180^\circ$: $x + y = 180^\circ$.
• $z = 61^\circ$ (from the given value in the diagram).

Solution:

1. Since $x = z$ and $z = 61^\circ$, $x = 61^\circ$.
2. $x + y = 180^\circ$:
   $61 + y = 180$ $y = 119^\circ$
3. $z = 61^\circ$ (already given).

Final Values:

• $x = 61^\circ$
• $y = 119^\circ$
• $z = 61^\circ$

Would you like a step-by-step explanation?

5 Questions to Expand Understanding:

1. How do the properties of a parallelogram help in solving angle problems?
2. Why do opposite angles in a parallelogram have the same measure?
3. How can you verify your answers for supplementary angles?
4. What happens if one angle of a parallelogram is $90^\circ$?
5. How can these principles apply to real-world geometry problems?

Tip:

Always double-check relationships between angles (supplementary or equal) to avoid calculation errors.