To determine if line $a$ and line $b$ are parallel, perpendicular, or neither, we need to calculate the slopes of both lines using the points provided.

Line $a$:

Line $a$ passes through points $(8, 6)$ and $(4, 13)$. The formula for slope is: $m = \frac{y_2 - y_1}{x_2 - x_1}$ For line $a$: $m_a = \frac{13 - 6}{4 - 8} = \frac{7}{-4} = -\frac{7}{4}$

Line $b$:

Line $b$ passes through points $(5, 13)$ and $(9, 6)$. Using the slope formula again: $m_b = \frac{6 - 13}{9 - 5} = \frac{-7}{4} = -\frac{7}{4}$

Conclusion:

Both lines have the same slope $m_a = m_b = -\frac{7}{4}$, so line $a$ and line $b$ are parallel.

Would you like more details on how slopes determine parallelism and perpendicularity?

Here are 5 related questions to consider:

1. How do you calculate the slope between two points in general?
2. What conditions make two lines perpendicular?
3. Can lines with zero slope be parallel or perpendicular?
4. How can the slope-intercept form help in comparing lines?
5. What happens when two lines have undefined slopes?

Tip: For perpendicular lines, the product of their slopes should be $-1$.