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

Properties of a Parallelogram:

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

From the diagram:

• One angle is $93^\circ$.
• The opposite angle is $-7y + 2$, so we equate $-7y + 2 = 93$.
• Adjacent angles $93^\circ$ and $-4z - 5$ are supplementary, so $93 + (-4z - 5) = 180$.
• Opposite angles $-2x + 3$ and $-4z - 5$ are equal, so $-2x + 3 = -4z - 5$.

Step 1: Solve for $y$

$-7y + 2 = 93$ $-7y = 91$ $y = -13$

Step 2: Solve for $z$

$93 + (-4z - 5) = 180$ $93 - 4z - 5 = 180$ $88 - 4z = 180$ $-4z = 92$ $z = -23$

Step 3: Solve for $x$

Using $-2x + 3 = -4z - 5$: $-2x + 3 = -4(-23) - 5$ $-2x + 3 = 92 - 5$ $-2x + 3 = 87$ $-2x = 84$ $x = -42$

Final Values:

• $x = -42$
• $y = -13$
• $z = -23$

Would you like further details or clarifications? Here are 5 related questions to explore:

1. Why are opposite angles in a parallelogram always equal?
2. How do supplementary angles relate to parallelograms?
3. What happens to angle properties if this was not a parallelogram but a rhombus?
4. Can the values of $x$, $y$, or $z$ be fractions in other cases?
5. How do we prove that adjacent angles are always supplementary in parallelograms?

Tip: Always verify your angle sums for consistency to ensure calculations are correct!