To determine which lines are parallel or perpendicular, we calculate the slopes of the lines. Recall:

1. The slope of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $m = \frac{y_2 - y_1}{x_2 - x_1}$

2. Lines are parallel if their slopes are equal.

3. Lines are perpendicular if the product of their slopes is $-1$ (their slopes are negative reciprocals).

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1. Calculate slopes for each line:

Line $a$:

Points: $(0, 4)$ and $(4, 3)$
$m_a = \frac{3 - 4}{4 - 0} = \frac{-1}{4} = -\frac{1}{4}$

Line $b$:

Points: $(0, 1)$ and $(4, 0)$
$m_b = \frac{0 - 1}{4 - 0} = \frac{-1}{4} = -\frac{1}{4}$

Line $c$:

Points: $(2, 0)$ and $(4, 4)$
$m_c = \frac{4 - 0}{4 - 2} = \frac{4}{2} = 2$

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2. Check for parallel lines:

Lines are parallel if their slopes are the same.

• Line $a$ and Line $b$ both have slopes $m = -\frac{1}{4}$.
  So, $a$ and $b$ are parallel.

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3. Check for perpendicular lines:

Lines are perpendicular if the product of their slopes is $-1$.

• For $a$ and $c$:
  $m_a \cdot m_c = \left(-\frac{1}{4}\right) \cdot 2 = -\frac{2}{4} = -\frac{1}{2} \neq -1$ $a$ and $c$ are not perpendicular.

• For $b$ and $c$:
  $m_b \cdot m_c = \left(-\frac{1}{4}\right) \cdot 2 = -\frac{2}{4} = -\frac{1}{2} \neq -1$ $b$ and $c$ are not perpendicular.

• For $a$ and $b$:
  $m_a$ and $m_b$ are the same ($-\frac{1}{4}$), so they cannot be perpendicular.

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Final Answer:

1. Lines $a$ and $b$ are parallel, because they have the same slope $-\frac{1}{4}$.
2. None of the lines are perpendicular, because no pair of slopes has a product of $-1$.

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Do you want further details or have any questions?
Here are 5 related questions to expand this topic:

1. How do you determine the slope of a vertical line?
2. Can two lines be neither parallel nor perpendicular?
3. What does the slope of a line tell you about its steepness?
4. How do you find the equation of a line given its slope and a point on it?
5. How can you determine if three lines are concurrent?

Tip: Always remember that parallel lines never intersect, while perpendicular lines intersect at a right angle!