Question

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:

Step1: Recall slope - formula

The slope formula for a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$.
For line $PQ$ with points $(w,v)$ and $(x,z)$, the slope $m_1=\frac{z - v}{x - w}$.
For line $P'Q'$ with points $(w + a,v + b)$ and $(x + a,z + b)$, the slope $m_2=\frac{(z + b)-(v + b)}{(x + a)-(w + a)}$.

Step2: Simplify the slope of $P'Q'$

Simplify $m_2=\frac{(z + b)-(v + b)}{(x + a)-(w + a)}=\frac{z - v}{x - w}$.
Since the two lines are parallel, their slopes are equal, and the slope of both lines is $\frac{z - v}{x - w}$.

Answer:

$\frac{z - v}{x - w}$