https://file.aiquickdraw.com/mathbot/file/1726091859002kevdel2g.jpg

The problem shown in the image asks to find the values of \( x \) and \( z \) from a diagram of intersecting lines. The two key expressions given are \( 7x + 71^\circ \) and \( 12x + 33^\circ \), both related to vertical angles. In this situation, vertical angles are congruent, meaning:

\[
7x + 71 = 12x + 33
\]

Now, let's solve this equation for \( x \):

### Step 1: Set up the equation
\[
7x + 71 = 12x + 33
\]

### Step 2: Subtract \( 7x \) from both sides to isolate \( x \) on one side:
\[
71 = 5x + 33
\]

### Step 3: Subtract 33 from both sides:
\[
71 - 33 = 5x
\]
\[
38 = 5x
\]

### Step 4: Divide by 5 to solve for \( x \):
\[
x = \frac{38}{5} = 7.6
\]

Now that we have \( x \), we can substitute it into one of the original expressions (e.g., \( 7x + 71 \)) to find \( z^\circ \).

### Step 5: Substitute \( x = 7.6 \) into \( 7x + 71 \):
\[
7(7.6) + 71 = 53.2 + 71 = 124.2^\circ
\]

Since \( z^\circ \) is the angle vertically opposite this, \( z = 124.2^\circ \).

### Final answers:
- \( x = 7.6 \)
- \( z = 124.2^\circ \)

Would you like any further explanation on any step?

### Related questions:
1. What are vertical angles, and why are they congruent?
2. How would the problem change if supplementary angles were involved instead?
3. How can we verify the answer for \( z \) in such angle problems?
4. What types of problems involve solving for variables within geometric figures like this one?
5. How do angle relationships differ in parallel line transversals compared to intersecting lines?

### Tip:  
Always double-check angle relationships (e.g., vertical, supplementary) when solving for variables in geometry problems.

Given the figure below, find the values of x and z: 7x + 71° and 12x + 33° are expressions for two vertical angles formed by intersecting lines.

This problem involves finding the values of x and z in an intersecting lines geometry problem using vertical angle relationships. Learn how to solve for x using algebraic methods, and then calculate z based on vertical angle congruence.

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