To determine how the line segments $\overline{AB}$ and $\overline{CD}$ are related, we need to calculate their slopes. If the slopes are equal, the lines are parallel. If the slopes are negative reciprocals of each other, the lines are perpendicular. If neither condition holds, they are neither parallel nor perpendicular.

Step 1: Find the slope of $\overline{AB}$

The formula for the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$\text{slope of } \overline{AB} = \frac{y_2 - y_1}{x_2 - x_1}$

For $A(3, 6)$ and $B(8, 7)$:

$\text{slope of } \overline{AB} = \frac{7 - 6}{8 - 3} = \frac{1}{5}$

Step 2: Find the slope of $\overline{CD}$

For $C(3, 3)$ and $D(8, 4)$:

$\text{slope of } \overline{CD} = \frac{4 - 3}{8 - 3} = \frac{1}{5}$

Step 3: Compare the slopes

Since the slope of $\overline{AB}$ is $\frac{1}{5}$ and the slope of $\overline{CD}$ is also $\frac{1}{5}$, the lines are parallel.

Thus, the correct answer is:

A. $\overline{AB} \parallel \overline{CD}$

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Would you like further details or have any questions? Here are 5 related questions:

1. How do we determine if two lines are perpendicular using slopes?
2. What does it mean for one line segment to bisect another?
3. How can we find the midpoint of a line segment?
4. Can parallel lines ever intersect, and under what conditions?
5. What is the equation of a line given a point and a slope?

Tip: When checking for parallelism, remember that equal slopes imply parallel lines.