Solve angle relationships to navigate the maze and find the value of x.

To solve the "Angle Relationships Maze" and find the value of \( x \), we need to navigate through the maze by solving each angle relationship problem step by step. The goal is to start at the "Start" box and reach the "Finish" box, using valid paths based on correct solutions.

Step-by-Step Solution:

#### Start Box:
- Given: \( 180^\circ - x \)
- Since the angles are supplementary (they form a straight line):
\[
180^\circ - x = 180^\circ
\]
\[
x = 0
\]
- Path: Move to the box with the number 40.

#### Box 40:
- Given: \( 125^\circ - x \)
- Since the angles are supplementary:
\[
125^\circ + x = 180^\circ
\]
\[
x = 180^\circ - 125^\circ
\]
\[
x = 55
\]
- Path: Move to the box with the number 55.

#### Box 55:
- Given: \( 58^\circ - x \)
- Since the angles are vertical angles:
\[
x = 58^\circ
\]
- Path: Move to the box with the number 58.

#### Box 58:
- Given: \( 2x + 46^\circ \)
- Since the angles are complementary (they form a right angle):
\[
2x + 46^\circ = 90^\circ
\]
\[
2x = 90^\circ - 46^\circ
\]
\[
2x = 44^\circ
\]
\[
x = 22
\]
- Path: Move to the box with the number 22.

#### Box 22:
- Given: \( 3x + 1 \)
- Since the angles are vertical angles:
\[
3x + 1 = 65^\circ
\]
\[
3x = 65^\circ - 1
\]
\[
3x = 64^\circ
\]
\[
x = \frac{64^\circ}{3}
\]
\[
x \approx 21.33
\]
- Path: Move to the box with the number 38.

#### Box 38:
- Given: \( 4x \)
- Since the angles are supplementary:
\[
4x + 100^\circ = 180^\circ
\]
\[
4x = 180^\circ - 100^\circ
\]
\[
4x = 80^\circ
\]
\[
x = 20
\]
- Path: Move to the box with the number 20.

#### Box 20:
- Given: \( 5x \)
- Since the angles are supplementary:
\[
5x + 55^\circ = 180^\circ
\]
\[
5x = 180^\circ - 55^\circ
\]
\[
5x = 125^\circ
\]
\[
x = 25
\]
- Path: Move to the box with the number 11.

#### Box 11:
- Given: \( 12x - 5^\circ \)
- Since the angles are vertical angles:
\[
12x - 5^\circ = 6x + 5^\circ
\]
\[
12x - 6x = 5^\circ + 5^\circ
\]
\[
6x = 10^\circ
\]
\[
x = \frac{10^\circ}{6}
\]
\[
x = \frac{5}{3}
\]
- Path: Move to the box with the number 5.

#### Box 5:
- Given: \( 6x - 5^\circ \)
- Since the angles are vertical angles:
\[
6x - 5^\circ = 5^\circ
\]
\[
6x = 5^\circ + 5^\circ
\]
\[
6x = 10^\circ
\]
\[
x = \frac{10^\circ}{6}
\]
\[
x = \frac{5}{3}
\]
- Path: Move to the "Finish" box.

Final Answer:

\[
\boxed{\frac{5}{3}}
\]