https://file.aiquickdraw.com/mathbot/file/1729551151136s1xjv0l6.jpg

In the given image, we are working with a rhombus, which has the following properties:

1. **Opposite angles are equal**.
2. **Adjacent angles are supplementary**, meaning they add up to \(180^\circ\).

### Analyzing the image:
- One angle is labeled as \(72^\circ\).
- The other three angles are expressed as algebraic terms involving \(x\), \(y\), and \(z\):
  - One angle: \((5x + 7)^\circ\)
  - Another angle: \((-y - 1)^\circ\)
  - Another angle: \((-z - 7)^\circ\)

### Step 1: Opposite angles in a rhombus
- The angle that is opposite the \(72^\circ\) angle is \((-y - 1)^\circ\).
  Thus:
  \[
  -y - 1 = 72 \quad \Rightarrow \quad y = -73
  \]

### Step 2: Adjacent angles are supplementary
- The angle adjacent to \(72^\circ\) is \((5x + 7)^\circ\).
  Since adjacent angles in a rhombus add up to \(180^\circ\):
  \[
  72 + (5x + 7) = 180
  \]
  Simplifying:
  \[
  5x + 79 = 180 \quad \Rightarrow \quad 5x = 101 \quad \Rightarrow \quad x = \frac{101}{5} = 20.2
  \]

### Step 3: The final angle relation
- The angle adjacent to \((5x + 7)^\circ\) is \((-z - 7)^\circ\).
  Since these angles are also supplementary:
  \[
  (5x + 7) + (-z - 7) = 180
  \]
  Substituting \(x = 20.2\) into the equation:
  \[
  (5(20.2) + 7) + (-z - 7) = 180
  \]
  Simplifying:
  \[
  (101 + 7) - z - 7 = 180 \quad \Rightarrow \quad 101 - z = 180 \quad \Rightarrow \quad z = -79
  \]

### Final values:
- \(x = 20.2\)
- \(y = -73\)
- \(z = -79\)

Would you like further details or clarification on any of the steps?

Here are some related questions you might consider:
1. How can we prove that the adjacent angles of any rhombus are supplementary?
2. What are the general properties of a rhombus that help solve angle-related problems?
3. How does this method change if the shape were a parallelogram instead of a rhombus?
4. Can this approach be used to solve for side lengths if the diagonals were given?
5. How do the angle properties of a rhombus relate to its diagonals?

**Tip**: When solving geometric problems, always keep track of symmetry or key properties like congruent and supplementary angles, as they simplify the process.

Find the value of x, y, and z in the rhombus below, where one angle is labeled as 72°, and the other three angles are expressed as algebraic terms: (5x + 7)°, (-y - 1)°, (-z - 7)°.

Learn how to find the values of x, y, and z in a rhombus using properties of geometry and algebra. This problem involves supplementary and congruent angles, and it provides step-by-step calculations for students in Grades 9-10.

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