The image you've uploaded is asking to find the slopes of two lines, $PQ$ and $RS$, and then determine whether the lines are parallel, perpendicular, or neither.

Here’s the step-by-step process for solving this:

1. Find the slope of line $PQ$

The coordinates for $P$ are $(-6, 16)$ and for $Q$ are $(2, -4)$. The formula for the slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

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

Substitute the values of the points:

$m_{PQ} = \frac{-4 - 16}{2 - (-6)} = \frac{-20}{8} = -\frac{5}{2}$

Thus, the slope of $PQ$ is $-\frac{5}{2}$.

2. Find the slope of line $RS$

The coordinates for $R$ are $(-6, 14)$ and for $S$ are $(-2, 4)$. Using the same formula for the slope:

$m_{RS} = \frac{4 - 14}{-2 - (-6)} = \frac{-10}{4} = -\frac{5}{2}$

Thus, the slope of $RS$ is also $-\frac{5}{2}$.

3. Compare the slopes

Both lines $PQ$ and $RS$ have the same slope, which means they are parallel.

Final Answers:

• Slope of $PQ$ = $-\frac{5}{2}$
• Slope of $RS$ = $-\frac{5}{2}$
• Since the slopes are equal, the lines are parallel.

Type: "par" for parallel.

Would you like further details on how to determine parallel or perpendicular lines? Let me know if you have any questions!

Here are some related questions to explore:

1. How do you determine if two lines are perpendicular?
2. What happens if two lines have slopes that are negative reciprocals of each other?
3. Can two lines be neither parallel nor perpendicular? How?
4. What is the equation of a line when you know the slope and one point?
5. How do the slopes of vertical and horizontal lines compare?

Tip: If two lines are parallel, their slopes will always be equal. If they are perpendicular, their slopes will be negative reciprocals of each other (i.e., their product will be $-1$).