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What are the values of x, y, and z? Use theorems to justify each answer. 

To find the values of \(x\), \(y\), and \(z\) using angle theorems, let's analyze the diagram. The diagram shows two intersecting lines forming several angles, with three of them marked as \(x^\circ\), \(y^\circ\), and \(z^\circ\). The given angles are \(74^\circ\) and \(66^\circ\).

### Step 1: Determine \(x\) using the Vertical Angles Theorem
- **Vertical Angles Theorem**: This theorem states that vertical (opposite) angles formed by the intersection of two lines are congruent.
- From the diagram, \(x\) and the angle labeled \(74^\circ\) are vertical angles.
- Therefore, \(x = 74^\circ\).

### Step 2: Determine \(y\) using the Linear Pair Theorem
- **Linear Pair Theorem**: This theorem states that if two angles form a linear pair (i.e., they are adjacent and their non-common sides are opposite rays), they are supplementary. This means their measures add up to \(180^\circ\).
- The angle \(y\) and the angle \(66^\circ\) form a linear pair.
- Therefore, \(y + 66^\circ = 180^\circ\).
- Solving for \(y\):
  \[
  y = 180^\circ - 66^\circ = 114^\circ.
  \]

### Step 3: Determine \(z\) using the Vertical Angles Theorem
- Similarly, \(z\) and \(66^\circ\) are vertical angles.
- By the Vertical Angles Theorem, \(z = 66^\circ\).

### Final Answer:
- \(x = 74^\circ\)
- \(y = 114^\circ\)
- \(z = 66^\circ\)

These values satisfy the conditions set by the relevant theorems.

Would you like further explanations or have any questions? Here are five related questions to consider:

1. How would the values change if the angles formed by the intersections were different?
2. Can you explain the difference between corresponding angles and vertical angles?
3. How does the Linear Pair Theorem apply to non-adjacent angles?
4. What other scenarios use the Vertical Angles Theorem in geometry?
5. Why is it important to distinguish between supplementary and complementary angles?

**Tip:** When solving angle problems involving intersecting lines, always consider using the Vertical Angles Theorem and Linear Pair Theorem as starting points.

Learn how to find the values of x, y, and z using angle theorems in geometry. This problem involves applying the Vertical Angles Theorem and Linear Pair Theorem to determine specific angles formed by intersecting lines. Suitable for students in Grades 7-9.

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