https://file.aiquickdraw.com/mathbot/file/1727226307687y9l0n89s.jpg

Let's analyze the geometry in the figure.

1. **Vertical Angles**: The given angles of 70° and \( (11x + 55)^\circ \) are **vertical angles**, which means they are equal.
   - \( 70^\circ = (11x + 55)^\circ \)
   
2. **Opposite Angles**: The \( z^\circ \) angle and \( 70^\circ \) are **opposite angles**, so they are equal as well:
   - \( z = 70^\circ \)

### Solve for \( x \):

We can solve the equation for \( x \):
\[
70 = 11x + 55
\]

Subtract 55 from both sides:
\[
70 - 55 = 11x
\]
\[
15 = 11x
\]

Now, divide both sides by 11:
\[
x = \frac{15}{11} \approx 1.36
\]

### Final values:
- \( x \approx 1.36 \)
- \( z = 70^\circ \)

Would you like more detailed steps or explanations on any part?

Here are 5 follow-up questions to expand your understanding:
1. What are vertical angles and why are they equal?
2. How are opposite angles in intersecting lines related?
3. Can you apply this concept to find angles in a triangle?
4. What if \( x \) represented a different function; how would it change the result?
5. How would you verify that your solution is correct?

**Tip**: In problems involving intersecting lines, remember that vertical angles are always equal.

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

Learn how to find the values of x and z in a geometric problem involving vertical and opposite angles. This step-by-step solution utilizes the Vertical Angles Theorem and basic algebra. Suitable for students in Grades 6-8.

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