1. **State the problem:** We are given two line segments AB and CD lying on lines with equations $2x - 4y = 8$ and $4x + 2y = 8$ respectively. We need to determine the relationship between these segments based on their slopes. 2. **Find the slope of line AB:** Rewrite $2x - 4y = 8$ in slope-intercept form $y = mx + b$. $$2x - 4y = 8 \implies -4y = -2x + 8 \implies y = \frac{-2x + 8}{-4} = \frac{-2x}{-4} + \frac{8}{-4} = \frac{1}{2}x - 2$$ So, the slope of line AB is $m_{AB} = \frac{1}{2}$. 3. **Find the slope of line CD:** Rewrite $4x + 2y = 8$ in slope-intercept form. $$4x + 2y = 8 \implies 2y = -4x + 8 \implies y = \frac{-4x + 8}{2} = -2x + 4$$ So, the slope of line CD is $m_{CD} = -2$. 4. **Analyze the slopes:** The slope of AB is $\frac{1}{2}$ and the slope of CD is $-2$. Two lines are perpendicular if their slopes are negative reciprocals, i.e., $m_1 = -\frac{1}{m_2}$. Check: $$-\frac{1}{m_{AB}} = -\frac{1}{\frac{1}{2}} = -2 = m_{CD}$$ This confirms the lines are perpendicular. 5. **Conclusion:** The correct statement is: "They are perpendicular because they have slopes that are opposite reciprocals of $-2$ and $\frac{1}{2}$." **Final answer:** They are perpendicular because their slopes are opposite reciprocals.