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QUAD is a quadrilateral with vertices [tex]\( Q(-3,2), U(3,0), A(6,-5), \)[/tex] and [tex]\( D(0,-3) \)[/tex].

The slope for [tex]\(\overline{QU}\)[/tex] is [tex]\(\frac{0-2}{3-(-3)} = -\frac{1}{3}\)[/tex].

The slope for [tex]\(\overline{UA}\)[/tex] is [tex]\(\frac{-5-0}{6-3} = -\frac{5}{3}\)[/tex].

The slope for [tex]\(\overline{AD}\)[/tex] is [tex]\(\frac{-3-(-5)}{0-6} = -\frac{1}{3}\)[/tex].

The slope for [tex]\(\overline{DQ}\)[/tex] is [tex]\(\frac{-3-2}{0-(-3)} = -\frac{5}{3}\)[/tex].

So, [tex]\(\quad\)[/tex] .

Therefore, QUAD is a parallelogram.

What is the missing step in the proof?

A. [tex]\(\overline{QU} \perp \overline{AD}\)[/tex] and [tex]\(\overline{UA} \perp \overline{DQ}\)[/tex] because the product of the slopes is [tex]\(-1\)[/tex].

B. [tex]\(\overline{QU} \parallel \overline{AD}\)[/tex] and [tex]\(\overline{UA} \parallel \overline{DQ}\)[/tex] because the segments have the same slope.

C. [tex]\(\overline{QU} \parallel \overline{AD}\)[/tex] and [tex]\(\overline{UA} \parallel \overline{DQ}\)[/tex] because the product of the slopes is [tex]\(-1\)[/tex].

D. [tex]\(\overline{QU} \perp \overline{AD}\)[/tex] and [tex]\(\overline{UA} \perp \overline{DQ}\)[/tex] because the segments have the same slope.



Answer :

GinnyAnswer
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.