To determine the angle between the two lines, we first need to calculate the slopes of each line.

Step 1: Calculate the slope of Line A

The slope of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

For Line A, which passes through the points $(-2, -7)$ and $(1, -9)$:

$m_A = \frac{-9 - (-7)}{1 - (-2)} = \frac{-9 + 7}{1 + 2} = \frac{-2}{3}$

So, the slope of Line A, $m_A$, is $-\frac{2}{3}$.

Step 2: Calculate the slope of Line B

For Line B, which passes through the points $(5, -1)$ and $(8, 3)$:

$m_B = \frac{3 - (-1)}{8 - 5} = \frac{3 + 1}{3} = \frac{4}{3}$

So, the slope of Line B, $m_B$, is $\frac{4}{3}$.

Step 3: Use the formula for the angle between two lines

The angle $\theta$ between two lines with slopes $m_1$ and $m_2$ is given by:

$\tan(\theta) = \left|\frac{m_1 - m_2}{1 + m_1m_2}\right|$

Substituting $m_A = -\frac{2}{3}$ and $m_B = \frac{4}{3}$:

$\tan(\theta) = \left|\frac{-\frac{2}{3} - \frac{4}{3}}{1 + \left(-\frac{2}{3}\right)\left(\frac{4}{3}\right)}\right|$

Simplifying the expression:

$\tan(\theta) = \left|\frac{-\frac{6}{3}}{1 - \frac{8}{9}}\right| = \left|\frac{-2}{\frac{1}{9}}\right| = \left|-\frac{18}{1}\right| = 18$

Step 4: Calculate the angle $\theta$

Now, we take the arctan (inverse tangent) to find the angle:

$\theta = \arctan(18)$

Using a calculator:

$\theta \approx 86.82^\circ$

Final Answer:

The angle between the two lines is approximately 86.82 degrees.

Would you like more details on any part of the solution?

Here are five related questions you might find interesting:

1. How do you calculate the angle between two lines that are perpendicular?
2. What is the significance of the angle between two intersecting lines?
3. How do you find the slope of a line parallel to a given line?
4. What is the equation of a line that passes through a given point and has a specific slope?
5. How does changing one point on a line affect the slope and angle?

Tip: When dealing with angles between lines, a small change in slope can lead to a large change in the resulting angle.