Determine the angle between the lines passing through the given points, round your answer to the nearest hundredth if necessary:

Line A passes through points: (-2, -7) and (1, -9)
Line B passes through points: (5, -1) and (8, 3)

To find the angle between two lines, we can use the formula involving the slopes of the two lines:

\[
\theta = \arctan \left( \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \right)
\]

where \( m_1 \) and \( m_2 \) are the slopes of lines A and B, respectively.

### Step 1: Calculate the slopes of the lines.

#### Slope of Line A:
The slope formula is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

For Line A (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 is \( m_A = -\frac{2}{3} \).

#### Slope of Line B:
For Line B (points \((5, -1)\) and \((8, 3)\)):

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

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

### Step 2: Use the angle formula.

Now, substitute the values of \( m_1 = -\frac{2}{3} \) and \( m_2 = \frac{4}{3} \) into the formula for the angle between the two lines:

\[
\theta = \arctan \left( \left| \frac{-\frac{2}{3} - \frac{4}{3}}{1 + \left(-\frac{2}{3}\right) \cdot \frac{4}{3}} \right| \right)
\]

Simplifying the expression inside the absolute value:

\[
\theta = \arctan \left( \left| \frac{-\frac{6}{3}}{1 + \left(-\frac{8}{9}\right)} \right| \right)
= \arctan \left( \left| \frac{-2}{1 - \frac{8}{9}} \right| \right)
= \arctan \left( \left| \frac{-2}{\frac{1}{9}} \right| \right)
= \arctan \left( \left| -18 \right| \right)
= \arctan(18)
\]

### Step 3: Calculate the angle.
Now, calculate \( \arctan(18) \):

\[
\theta \approx 86.82^\circ
\]

Thus, the angle between the two lines is approximately \( 86.82^\circ \).

---

Do you have any questions or would you like more details?

Here are five related questions:
1. How do you find the slope of a line given two points?
2. What is the geometrical meaning of the slope of a line?
3. How do you interpret the angle between two lines?
4. What happens if the slopes of two lines are the same?
5. How do perpendicular lines relate to the angle between two lines?

**Tip:** If two lines are perpendicular, their slopes satisfy \( m_1 \cdot m_2 = -1 \).

Determine the angle between the lines passing through the given points, round your answer to the nearest hundredth if necessary: Line A passes through points: (-2, -7) and (1, -9) Line B passes through points: (5, -1) and (8, 3)

Calculate the angle between two lines passing through the points (-2, -7), (1, -9) and (5, -1), (8, 3). This problem involves key concepts in algebra and analytic geometry, including slope calculations and finding angles between lines, suitable for students in Grades 9-12.

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