The problem involves a parallelogram and requires solving for the values of $x$, $y$, and $z$. Let's solve step by step using the properties of parallelograms:

Key properties of a parallelogram:

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

The given angles are:

• $67^\circ$
• $-y + 7^\circ$
• $10x - 3^\circ$
• $-4z + 5^\circ$.

Step 1: Solve for $y$

The angles $67^\circ$ and $-y + 7^\circ$ are consecutive angles. By the property of parallelograms, they must sum to $180^\circ$: $67 + (-y + 7) = 180.$ Simplify: $67 + 7 - y = 180 \quad \Rightarrow \quad 74 - y = 180 \quad \Rightarrow \quad y = -106.$

Step 2: Solve for $x$

The angles $67^\circ$ and $10x - 3^\circ$ are opposite angles. Opposite angles in a parallelogram are equal: $10x - 3 = 67.$ Solve for $x$: $10x = 70 \quad \Rightarrow \quad x = 7.$

Step 3: Solve for $z$

The angles $-4z + 5^\circ$ and $67^\circ$ are consecutive angles. They must sum to $180^\circ$: $-4z + 5 + 67 = 180.$ Simplify: $-4z + 72 = 180 \quad \Rightarrow \quad -4z = 108 \quad \Rightarrow \quad z = -27.$

Final Answers:

$x = 7, \quad y = -106, \quad z = -27.$

Do you need further clarification on any step, or help with similar problems?

Here are 5 related questions for practice:

1. If opposite angles in a parallelogram are $3x + 5^\circ$ and $2x + 15^\circ$, what is the value of $x$?
2. A parallelogram has one angle $45^\circ$. What are the other angles?
3. What is the sum of all interior angles in a parallelogram?
4. In a parallelogram, one angle is $90^\circ$. What type of parallelogram is it?
5. How do the diagonals of a parallelogram relate to each other in terms of their properties?

Tip: Always check both the supplementary and equality properties when working with parallelogram angles!