We are tasked with finding the smallest positive angle between two lines, $L_1$ and $L_2$, with given points.

Step 1: Find the slopes of the lines.

1. Slope of $L_1$:
   The points on $L_1$ are $(-4, 2)$ and $(-4, -1)$.
   The slope formula is: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Since the x-coordinates are the same ($x_1 = x_2 = -4$), the line is vertical, so the slope is undefined.

2. Slope of $L_2$:
   The points on $L_2$ are $(0, -5)$ and $(2, 1)$.
   Using the slope formula: $m = \frac{1 - (-5)}{2 - 0} = \frac{6}{2} = 3$ So, the slope of $L_2$ is $m_2 = 3$.

Step 2: Calculate the angle between the lines.

To find the angle $\theta$ between two lines, we use the formula: $\tan(\theta) = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right|$ Where $m_1$ is the slope of the first line, and $m_2$ is the slope of the second line.

Since $L_1$ is vertical, its slope $m_1$ is undefined, which means the angle between $L_1$ and $L_2$ is a right angle. The angle between a vertical line and any non-vertical line is always $90^\circ$.

Thus, the smallest positive angle between $L_1$ and $L_2$ is 90 degrees.

Summary:

The smallest positive angle from $L_1$ to $L_2$ is 90 degrees.

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Would you like a more detailed breakdown of any part of this solution? Here are some related questions you might find helpful:

1. How do you calculate the slope of a line using two points?
2. What does it mean when a line has an undefined slope?
3. How can you find the angle between two non-parallel lines?
4. What are the properties of a vertical line?
5. How does the formula for the angle between two lines work for non-vertical lines?

Tip: When you have a vertical line, remember that its slope is undefined, and it will always form a right angle (90°) with any line that isn't vertical.