In the given parallelogram, we are tasked with finding the values of the angles $x^\circ$, $y^\circ$, and $z^\circ$. Let's break it down step by step:

Key properties of a parallelogram:

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

Analyzing the angles:

• We are given two adjacent angles: $42^\circ$ and $46^\circ$.

Step 1: Find $z^\circ$

Since $z^\circ$ is opposite to $46^\circ$ in the parallelogram, they must be equal: $z = 46^\circ$

Step 2: Find $y^\circ$

Next, we know that adjacent angles in a parallelogram add up to $180^\circ$. Therefore, to find $y^\circ$, which is adjacent to $42^\circ$: $y + 42 = 180^\circ$ Solving for $y$: $y = 180^\circ - 42^\circ = 138^\circ$

Step 3: Find $x^\circ$

Since $x^\circ$ is opposite to $y^\circ$ in the parallelogram, and opposite angles are equal: $x = y = 138^\circ$

Final values:

• $x = 138^\circ$
• $y = 138^\circ$
• $z = 46^\circ$

Would you like further details or clarifications?

Here are 5 follow-up questions to expand on this information:

1. What are the properties of adjacent angles in other quadrilaterals like rectangles or trapezoids?
2. How would the angle relationships change if this were a rhombus instead of a parallelogram?
3. Can we calculate the sides of this parallelogram given more information?
4. How can we use the interior angle sum theorem in polygons with more sides?
5. What are the special cases of parallelograms like squares and rectangles?

Tip: Always remember that in any parallelogram, opposite angles are equal and adjacent angles sum to $180^\circ$. This is a fundamental property used frequently in geometry.