Let’s analyze the problem.

Key Observations:

1. Parallelogram Properties: In a parallelogram:

   • Opposite angles are equal.
   • Adjacent angles are supplementary (sum to $180^\circ$).

2. Angles Given:

   • One angle is $79^\circ$.
   • Adjacent angles include $5y - 4$ and $x + 5$.
   • Opposite the $79^\circ$ angle is $9z - 7$.

Steps to Solve:

1. Equate Opposite Angles:

   • Since opposite angles are equal: $9z - 7 = 79$ Solve for $z$: $9z = 86 \implies z = \frac{86}{9} \approx 9.56.$

2. Set Adjacent Angles as Supplementary:

   • Adjacent angles $79^\circ$ and $5y - 4$ must add to $180^\circ$: $79 + (5y - 4) = 180$ Simplify: $5y + 75 = 180 \implies 5y = 105 \implies y = 21.$

3. Relate $x + 5$:

   • Adjacent angles $x + 5$ and $5y - 4$ are supplementary: $(x + 5) + (5y - 4) = 180.$ Substitute $y = 21$: $(x + 5) + (5(21) - 4) = 180$ Simplify: $(x + 5) + 101 = 180 \implies x + 106 = 180 \implies x = 74.$

Final Answer:

$x = 74, \quad y = 21, \quad z = 9.56.$

Let me know if you'd like detailed explanations or verification.

5 Related Questions:

1. What are the conditions for a quadrilateral to be a parallelogram?
2. How do you verify angle properties in other types of quadrilaterals, like rhombuses or rectangles?
3. What happens if all angles of a parallelogram are equal?
4. How do supplementary angles relate to cyclic quadrilaterals?
5. Can $x, y,$ and $z$ take fractional values in this problem? Why?

Tip:

Always confirm the relationship between adjacent and opposite angles when working with parallelograms to avoid calculation errors.