https://r2.erweima.ai/mathbot/image/1726397220442iiexviyn.jpg

To solve for the values of \(x\) and \(z\) in the given figure, we need to rely on properties of intersecting lines and angles.

1. **Vertical angles**: Vertical angles are equal. Therefore, the angle labeled \(97^\circ\) is equal to the angle labeled \( (14x - 71)^\circ \).
   
   This gives the equation:
   \[
   14x - 71 = 97
   \]

2. **Linear pair**: The angles \(97^\circ\) and \(z^\circ\) form a linear pair, which means they are supplementary. That means their sum is \(180^\circ\). Hence:
   \[
   97 + z = 180
   \]
   Solving these two equations will give us the values of \(x\) and \(z\).

### Solving for \(x\):
\[
14x - 71 = 97
\]
Add 71 to both sides:
\[
14x = 168
\]
Divide both sides by 14:
\[
x = 12
\]

### Solving for \(z\):
\[
97 + z = 180
\]
Subtract 97 from both sides:
\[
z = 83
\]

Thus, the values are:
\[
x = 12, \quad z = 83
\]

Would you like more details on this, or do you have any further questions?

### Related Questions:
1. What are vertical angles, and why are they equal?
2. What are supplementary angles and how do they apply to intersecting lines?
3. Can you explain the difference between adjacent and vertical angles?
4. How do we solve equations involving linear pairs of angles?
5. What is the significance of solving for unknown angles in geometry?

### Tip:
In geometry, always look for angle relationships like vertical angles, corresponding angles, and supplementary angles to help solve problems involving unknown angles.

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

Learn how to solve for unknown angles x and z using vertical angles and linear pairs. This problem applies key geometric theorems and algebraic manipulation, suitable for students in Grades 7-9.

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