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To solve for \( x \), \( y \), and \( z \) in the given rhombus, we need to utilize the properties of a rhombus, especially the fact that opposite angles are equal.

### Key properties of a rhombus:
1. Opposite angles are equal.
2. The sum of the angles in any quadrilateral is \( 360^\circ \).

Given the following information:
- One angle is \( 71^\circ \).
- Another angle is \( (-6y + 1)^\circ \).
- A third angle is \( (-4z + 5)^\circ \).
- The fourth angle is \( (-x - 8)^\circ \).

### Step-by-step solution:

#### Step 1: Opposite angles are equal.
- The angle opposite the \( 71^\circ \) angle will also be \( 71^\circ \).
- Therefore, we can set:
  \[
  -4z + 5 = 71
  \]
  Solve for \( z \):
  \[
  -4z = 71 - 5
  \]
  \[
  -4z = 66
  \]
  \[
  z = \frac{-66}{-4} = 16.5
  \]

#### Step 2: Sum of angles in a rhombus is \( 360^\circ \).
The sum of all angles is:
\[
71 + (-x - 8) + (-6y + 1) + 71 = 360
\]
Simplify the equation:
\[
71 + (-x - 8) + (-6y + 1) + 71 = 360
\]
\[
142 - x - 8 - 6y + 1 = 360
\]
\[
135 - x - 6y = 360
\]
\[
-x - 6y = 360 - 135
\]
\[
-x - 6y = 225
\]

#### Step 3: Solve for \( x \) and \( y \).

We also know that opposite angles are equal:
\[
-6y + 1 = -x - 8
\]
Simplify:
\[
-6y + 1 = -x - 8
\]
\[
-6y + x = -9
\]

Now we have a system of equations:
1. \( -x - 6y = 225 \)
2. \( -6y + x = -9 \)

We can solve this system of equations by substitution or elimination. Let's solve it:

From equation (2), solve for \( x \):
\[
x = 6y - 9
\]

Substitute this into equation (1):
\[
-(6y - 9) - 6y = 225
\]
Simplify:
\[
-6y + 9 - 6y = 225
\]
\[
-12y + 9 = 225
\]
\[
-12y = 225 - 9
\]
\[
-12y = 216
\]
\[
y = \frac{216}{-12} = -18
\]

#### Step 4: Solve for \( x \).

Now, substitute \( y = -18 \) back into \( x = 6y - 9 \):
\[
x = 6(-18) - 9
\]
\[
x = -108 - 9 = -117
\]

### Final answers:
- \( x = -117 \)
- \( y = -18 \)
- \( z = 16.5 \)

Would you like more details or have any questions?

### Related Questions:
1. How do you find angles in a parallelogram if opposite angles are equal?
2. How does the sum of interior angles in a quadrilateral help in solving problems?
3. What are the properties of rhombuses and how do they differ from other quadrilaterals?
4. How do systems of equations help in solving geometric problems?
5. Can the method of substitution always be used to solve systems of equations in geometry?

**Tip:** When solving geometric problems, always check if the shape has symmetry or special properties, like equal angles or sides, to simplify your calculations.

Find the value of x, y, and z in the rhombus below.

Learn how to find the values of x, y, and z in a rhombus using geometric properties and algebraic equations. This problem covers key concepts such as opposite angles in a rhombus and solving systems of equations, suitable for students in Grades 8-10.

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