https://file.aiquickdraw.com/mathbot/file/17265296525103y3t9207.jpg

The image shows two intersecting lines forming vertical angles, with the angles labeled as follows:

- \( z^\circ \)
- \( (8x + 44)^\circ \)
- \( (5x + 71)^\circ \)

From geometry, we know two things:
1. **Vertical angles are equal**.
   - So, \( z^\circ = (8x + 44)^\circ \) and \( (5x + 71)^\circ = (8x + 44)^\circ \).
   
2. **Adjacent angles are supplementary**, meaning they sum to 180°.

Let's solve this step-by-step.

### Step 1: Set up the equation for the adjacent angles:
\[
(5x + 71)^\circ + (8x + 44)^\circ = 180^\circ
\]

### Step 2: Solve for \( x \):

\[
5x + 71 + 8x + 44 = 180
\]
\[
13x + 115 = 180
\]
\[
13x = 180 - 115
\]
\[
13x = 65
\]
\[
x = \frac{65}{13} = 5
\]

### Step 3: Find \( z \):
Since \( z = (8x + 44)^\circ \), substitute \( x = 5 \) into the equation:
\[
z = 8(5) + 44 = 40 + 44 = 84^\circ
\]

### Final Answers:
- \( x = 5 \)
- \( z = 84^\circ \)

Would you like a more detailed explanation, or do you have any questions?

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Here are 5 related questions to explore:
1. How would the solution change if the given angles were part of a triangle?
2. What if the lines were parallel—how would alternate interior angles apply?
3. Can you find the value of \( x \) if the adjacent angles were instead equal?
4. How are supplementary and complementary angles different?
5. What geometric properties help us determine the sum of angles in any polygon?

**Tip:** Vertical angles are always congruent, making them a key property in many geometry problems!

Given the figure, find the values of x and z based on the angles (5x + 71)°, (8x + 44)°, and z° in intersecting lines.

Learn how to find the values of x and z in an intersecting lines geometry problem. This problem involves algebraic expressions for vertical and supplementary angles, applying the Vertical Angles Theorem and Supplementary Angles Theorem. Suitable for Grades 8-10.

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