https://file.aiquickdraw.com/mathbot/file/1725472899282sdrw2vno.jpg

The image you uploaded shows a figure with intersecting lines and some angles labeled. The problem asks to find the values of \( x \) and \( z \).

### Explanation:

Given:
- The angle opposite to \( 78^\circ \) is also \( 78^\circ \) (vertical angles are equal).
- The angle opposite to \( z^\circ \) is \( z^\circ \) (vertical angles are equal).
- The angle \( (10x + 58)^\circ \) is adjacent to \( 78^\circ \) and \( z^\circ \).

### Steps to solve:

1. **Use the property of vertical angles:**
   \[
   \text{Angle opposite } 78^\circ = 78^\circ
   \]
   \[
   \text{Angle opposite } z^\circ = z^\circ
   \]

2. **Use the linear pair (angles on a straight line) property:**
   The sum of adjacent angles on a straight line is \( 180^\circ \).
   \[
   78^\circ + (10x + 58)^\circ = 180^\circ
   \]

3. **Solve the equation for \( x \):**
   \[
   78 + 10x + 58 = 180
   \]
   Combine like terms:
   \[
   10x + 136 = 180
   \]
   Subtract 136 from both sides:
   \[
   10x = 44
   \]
   Divide by 10:
   \[
   x = 4.4
   \]

4. **Determine \( z \):**
   Since \( z \) is vertical to \( 78^\circ \), and these are vertical angles:
   \[
   z = 78^\circ
   \]

### Final Answers:
- \( x = 4.4 \)
- \( z = 78^\circ \)

Would you like a more detailed explanation or have any questions about these steps? 

Here are five related questions that could further your understanding:

1. What is the linear pair of angles and how does it apply to this problem?
2. How do you determine vertical angles, and why are they equal?
3. Can \( x \) take any value other than \( 4.4 \) to satisfy the equation?
4. What are supplementary angles, and how do they differ from complementary angles?
5. How would the solution change if the angle opposite \( z^\circ \) was given as a different value?

**Tip:** Remember, vertical angles are always equal, and angles on a straight line always add up to 180°.

Given the figure below, find the values of x and z.

This problem helps students find the values of x and z using the concepts of vertical angles and linear pair angles. The solution includes a detailed step-by-step explanation suitable for students in Grades 8-10.

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