Here is the solution to the problems based on the given image:

Problems 20-22: Parallel Slope

To find the slope of a line parallel to the line passing through each pair of points, we calculate the slope using the formula:

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

• 20) Points: (4, 0) and (1, 7) $m = \frac{7 - 0}{1 - 4} = \frac{7}{-3} = -\frac{7}{3}$

  The slope of a parallel line is also $-\frac{7}{3}$.

• 21) Points: (-3, -4) and (1, -5) $m = \frac{-5 - (-4)}{1 - (-3)} = \frac{-5 + 4}{1 + 3} = \frac{-1}{4}$

  The slope of a parallel line is $-\frac{1}{4}$.

• 22) Points: (-5, -6) and (-2, -3) $m = \frac{-3 - (-6)}{-2 - (-5)} = \frac{-3 + 6}{-2 + 5} = \frac{3}{3} = 1$

  The slope of a parallel line is $1$.

Problems 23-25: Perpendicular Slope

For perpendicular lines, the slope is the negative reciprocal of the original slope:

$m_{\text{perpendicular}} = -\frac{1}{m}$

• 23) Points: (-4, 4) and (6, 3) $m = \frac{3 - 4}{6 - (-4)} = \frac{-1}{10}$ $m_{\text{perpendicular}} = -\frac{1}{\left(-\frac{1}{10}\right)} = 10$

• 24) Points: (5, -1) and (0, 1) $m = \frac{1 - (-1)}{0 - 5} = \frac{1 + 1}{-5} = \frac{2}{-5} = -\frac{2}{5}$ $m_{\text{perpendicular}} = -\frac{1}{\left(-\frac{2}{5}\right)} = \frac{5}{2}$

• 25) Points: (8, -6) and (1, -1) $m = \frac{-1 - (-6)}{1 - 8} = \frac{-1 + 6}{-7} = \frac{5}{-7} = -\frac{5}{7}$ $m_{\text{perpendicular}} = -\frac{1}{\left(-\frac{5}{7}\right)} = \frac{7}{5}$

Problems 26-27: Parallel Lines

Using the figure provided:

• 26) If $\angle 1$ is congruent to $\angle 5$, which two lines are parallel? Corresponding angles are congruent, so lines $c$ and $d$ are parallel.

• 27) If $\angle 2$ is congruent to $\angle 7$, which two lines are parallel? Alternate interior angles are congruent, so lines $a$ and $b$ are parallel.

Questions to Explore Further:

1. Why do parallel lines have the same slope?
2. How is the perpendicular slope derived mathematically?
3. What other geometric properties help identify parallel lines?
4. Can you identify slopes directly from a graph instead of calculation?
5. How do slopes change when dealing with vertical or horizontal lines?

Tip: Remember, perpendicular slopes are always negative reciprocals, so double-check signs! Let me know if you'd like more details!