Solution:

We are given a parallelogram $ABCD$, and we need to determine the measure of $x^\circ$. Here are the properties of parallelograms that are relevant to this problem:

1. Opposite angles are equal.
2. Adjacent angles are supplementary (their sum is $180^\circ$).

Step 1: Relating $x^\circ$ and the angle at $A$:

The angle at $A$ is $65^\circ$. Since adjacent angles in a parallelogram are supplementary: $x^\circ + 65^\circ = 180^\circ$

Step 2: Solve for $x^\circ$:

$x^\circ = 180^\circ - 65^\circ = 115^\circ$

Thus, the measure of $x^\circ$ is: $\boxed{115^\circ}$

Would you like a detailed explanation of the other angles ($y^\circ$ and $z^\circ$) or any further assistance?

Related Questions:

1. What is the value of $y^\circ$ in the parallelogram, and how can it be calculated?
2. What are the properties of opposite angles in parallelograms, and how do they relate to this problem?
3. If one angle in a parallelogram is $50^\circ$, what are the measures of all the other angles?
4. How can you prove that adjacent angles in a parallelogram are supplementary using geometry principles?
5. What changes if this shape were a rectangle or rhombus instead of a generic parallelogram?

Tip:

Always use the property of supplementary adjacent angles in parallelograms when given one angle to determine the others quickly!