Question

determine the missing information in the paragraph proof.
given: line pq contains points (w, v) and (x, z) and line p’q’ contains points (w + a, v + b) and (x + a, z + b). lines pq and p’q’ are parallel.
prove: parallel lines have the same slope.
image of two lines with points q’(w + a, v + b), q(w, v), p’(x + a, z + b), p(x, z)
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.
options:

• ( \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:

Step1: Calculate slope of PQ

For line PQ with points \((w, v)\) and \((x, z)\), using slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), we get \(m_{PQ}=\frac{z - v}{x - w}\).

Step2: Calculate slope of P'Q'

For line P'Q' with points \((w + a, v + b)\) and \((x + a, z + b)\), slope \(m_{P'Q'}=\frac{(z + b)-(v + b)}{(x + a)-(w + a)}=\frac{z - v}{x - w}\).
So the slope of both lines is \(\frac{z - v}{x - w}\).

Answer:

A. \(\frac{z - v}{x - w}\)