To determine if QUAD is a parallelogram, we need to use the properties of the slopes of its sides. The given information includes the slopes of the four sides of quadrilateral QUAD:
- The slope of [tex]\(\overline{QU}\)[/tex] is [tex]\(-\frac{1}{3}\)[/tex].
- The slope of [tex]\(\overline{UA}\)[/tex] is [tex]\(-\frac{5}{3}\)[/tex].
- The slope of [tex]\(\overline{AD}\)[/tex] is [tex]\(-\frac{1}{3}\)[/tex].
- The slope of [tex]\(\overline{DQ}\)[/tex] is [tex]\(-\frac{5}{3}\)[/tex].
To prove QUAD is a parallelogram, we need to show that both pairs of opposite sides are parallel. In geometry, two lines are parallel if they have the same slope.
1. Check Opposite Sides:
- For [tex]\(\overline{QU}\)[/tex] and [tex]\(\overline{AD}\)[/tex]:
[tex]\[
\text{Slope of } \overline{QU} = -\frac{1}{3} \quad \text{and} \quad \text{Slope of } \overline{AD} = -\frac{1}{3}.
\][/tex]
Since the slopes are equal, [tex]\(\overline{QU} \parallel \overline{AD}\)[/tex].
- For [tex]\(\overline{UA}\)[/tex] and [tex]\(\overline{DQ}\)[/tex]:
[tex]\[
\text{Slope of } \overline{UA} = -\frac{5}{3} \quad \text{and} \quad \text{Slope of } \overline{DQ} = -\frac{5}{3}.
\][/tex]
Since the slopes are equal, [tex]\(\overline{UA} \parallel \overline{DQ}\)[/tex].
2. Conclusion:
Since each pair of opposite sides of the quadrilateral has the same slope, we conclude that they are parallel.
Therefore, the correct step that completes the proof is:
B. [tex]\(\overline{QU} \parallel \overline{AD}\)[/tex] and [tex]\(\overline{UA} \parallel \overline{DQ}\)[/tex] because the segments have the same slope.
This reasoning shows that QUAD is indeed a parallelogram.
