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In this problem, we are given a parallelogram with angles expressed in terms of the variables \( x \), \( y \), and \( z \). We are tasked with finding the values of these variables. Let's break it down step by step.

### Key properties of a parallelogram:
1. Opposite angles are equal.
2. Adjacent angles are supplementary (i.e., they add up to \(180^\circ\)).

The angles provided are:
- \( 67^\circ \) (bottom-left)
- \( (-4z + 5)^\circ \) (bottom-right)
- \( (-y + 7)^\circ \) (top-left)
- \( (10x - 3)^\circ \) (top-right)

### Step 1: Use the property of opposite angles being equal.
For opposite angles:
- \( (-4z + 5)^\circ = 67^\circ \)
- \( (-y + 7)^\circ = (10x - 3)^\circ \)

Let's solve these one at a time.

### Step 2: Solve for \( z \)
From \( (-4z + 5) = 67 \):
\[
-4z + 5 = 67
\]
\[
-4z = 67 - 5
\]
\[
-4z = 62
\]
\[
z = -\frac{62}{4} = -15.5
\]

### Step 3: Solve for \( y \) and \( x \)
From \( (-y + 7) = (10x - 3) \):
\[
-y + 7 = 10x - 3
\]
Rearranging this equation to isolate \( y \) on one side:
\[
-y = 10x - 3 - 7
\]
\[
-y = 10x - 10
\]
\[
y = -10x + 10
\]
Now, we have \( y \) in terms of \( x \).

### Step 4: Use the property of supplementary adjacent angles.
The angles \( 67^\circ \) and \( (-y + 7)^\circ \) are adjacent, so they must add up to \( 180^\circ \):
\[
67 + (-y + 7) = 180
\]
\[
67 - y + 7 = 180
\]
\[
74 - y = 180
\]
\[
y = 74 - 180 = -106
\]

### Step 5: Solve for \( x \)
Since we know \( y = -106 \), substitute this into the equation \( y = -10x + 10 \):
\[
-106 = -10x + 10
\]
\[
-10x = -106 - 10
\]
\[
-10x = -116
\]
\[
x = \frac{-116}{-10} = 11.6
\]

### Final Results:
- \( x = 11.6 \)
- \( y = -106 \)
- \( z = -15.5 \)

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

Here are 5 questions to expand on this concept:
1. How can you check your solutions for \( x \), \( y \), and \( z \)?
2. What other properties of parallelograms could help in similar problems?
3. How would this approach change if the quadrilateral were a rectangle?
4. Can you generalize this method for any quadrilateral with known properties?
5. How do we use algebraic manipulation to verify supplementary angle relationships?

**Tip:** Always verify your final answers by checking if they satisfy all properties of the shape or equation you are solving for.

Find the value of x, y, and z in the parallelogram where the angles are expressed as (-y + 7)°, (10x - 3)°, 67°, and (-4z + 5)°.

Learn how to solve for the values of x, y, and z in a parallelogram where the angles are given by (-y + 7)°, (10x - 3)°, 67°, and (-4z + 5)°. This problem involves using key properties of parallelograms, such as opposite angles being equal and adjacent angles summing to 180°.

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