Question

determine the missing information in the paragraph proof.

given: line pq contains points \\((w, v)\\) and \\((x, z)\\) and line pq contains points \\((w + a, v + b)\\) and \\((x + a, z + b)\\). lines pq and pq are parallel.

prove: parallel lines have the same slope.

since slope is calculated using the formula \\(m = \frac{y_2 - y_1}{x_2 - x_1}\\), the slope of both lines is equivalent to ______. it is given that the lines are parallel, and we calculated that the slopes are the same. therefore, parallel lines have the same slopes.

• \\(\frac{z - v}{x - w}\\)
• \\(\frac{x - w}{z - v}\\)
• \\(\frac{v - z + b}{x - z + a}\\)
• \\(\frac{w - x + a}{v - z + b}\\)

Explanation:

Identify given coordinates

Using the Slope of a Line knowledge point
The line \(PQ\) contains the points \(Q(w, v)\) and \(P(x, z)\).

Calculate slope of line PQ

Using the Slope of a Line knowledge point

$$ m_{PQ} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{z - v}{x - w} $$

Calculate slope of line P'Q'

Using the Slope of a Line knowledge point
The line \(P'Q'\) contains the points \(Q'(w + a, v + b)\) and \(P'(x + a, z + b)\).

$$ m_{P'Q'} = \frac{(z + b) - (v + b)}{(x + a) - (w + a)} = \frac{z - v}{x - w} $$

Match with options

Using the Slope of a Line knowledge point
Both slopes simplify to \(\frac{z - v}{x - w}\).

Answer:

• (A) \(\frac{z - v}{x - w}\) (Correct answer)
• (B) \(\frac{x - w}{z - v}\)
• (C) \(\frac{v - z + b}{x - z + a}\)
• (D) \(\frac{w - x + a}{v - z + b}\)