To determine if lines $A$ and $B$ are parallel, perpendicular, or neither, we need to calculate the slopes of both lines and compare them.

Step 1: Calculate the slope of Line $A$

Line $A$ passes through the points $(0, 9)$ and $(6, 5)$. The formula for the slope $m$ of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

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

For Line $A$:

$m_A = \frac{5 - 9}{6 - 0} = \frac{-4}{6} = -\frac{2}{3}$

Step 2: Calculate the slope of Line $B$

Line $B$ passes through the points $(-6, -3)$ and $(0, -7)$.

For Line $B$:

$m_B = \frac{-7 - (-3)}{0 - (-6)} = \frac{-7 + 3}{0 + 6} = \frac{-4}{6} = -\frac{2}{3}$

Step 3: Compare the slopes

• Parallel: If the slopes of the two lines are equal, the lines are parallel.
• Perpendicular: If the product of the slopes is $-1$, the lines are perpendicular.
• Neither: If neither of the above conditions is met, the lines are neither parallel nor perpendicular.

Here, $m_A = -\frac{2}{3}$ and $m_B = -\frac{2}{3}$.

Since $m_A = m_B$, the lines are parallel.

Conclusion

The correct answer is:

Parallel

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Would you like more details or have any questions? Here are some related questions:

1. How do you find the slope of a line given two points?
2. What are the properties of parallel lines?
3. How do you determine if two lines are perpendicular?
4. Can two lines be parallel if their slopes are different?
5. What is the relationship between the slopes of perpendicular lines?

Tip: When two lines are parallel, their slopes are exactly the same.