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.
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} )
image shows points q(w, v), q(w+a, v+b), p(x, z), p(x+a, z+b) with two lines

Explanation:

Step1: Recall slope formula

The slope formula is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). We need to find the slope for both lines PQ and P'Q' and show they are equal.

Step2: Calculate slope of PQ

For line PQ with points \( (w, v) \) and \( (x, z) \), using the slope formula:
\( m_{PQ} = \frac{z - v}{x - w} \)

Step3: Calculate slope of P'Q'

For line P'Q' with points \( (w + a, v + b) \) and \( (x + a, z + b) \), apply the slope formula:
\( m_{P'Q'} = \frac{(z + b) - (v + b)}{(x + a) - (w + a)} \)
Simplify numerator and denominator:
Numerator: \( (z + b) - (v + b) = z + b - v - b = z - v \)
Denominator: \( (x + a) - (w + a) = x + a - w - a = x - w \)
So, \( m_{P'Q'} = \frac{z - v}{x - w} \)

Step4: Identify the equivalent slope

From the calculations, both slopes (of PQ and P'Q') are equal to \( \frac{z - v}{x - w} \).

Answer:

\(\frac{z - v}{x - w}\) (the first option)