Question

the slope criteria
reset
the slope of pq is (v - z) divided by (w - x)
the slope of pq is (v + b) - (z + b) divided by (w + a) - (z + a)
both lines have a slope of (v - z) divided by (w - x)
therefore, the lines are parallel

Explanation:

Step1: Calculate the slope of $\overleftrightarrow{PQ}$

The formula for the slope of 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 points $P(x,z)$ and $Q(w,v)$, the slope of $\overleftrightarrow{PQ}$ is $\frac{v - z}{w - x}$.

Step2: Calculate the slope of $\overleftrightarrow{P'Q'}$

For points $P'(x + a,z + b)$ and $Q'(w + a,v + b)$, using the slope formula $m=\frac{y_2 - y_1}{x_2 - x_1}$, we have $y_2=v + b,y_1=z + b,x_2=w + a,x_1=x + a$. The slope is $\frac{(v + b)-(z + b)}{(w + a)-(x + a)}$. Simplify the denominator: $(w + a)-(x + a)=w - x$, and the numerator: $(v + b)-(z + b)=v - z$.

Answer:

The slope of $\overleftrightarrow{P'Q'}$ is $(v - z)$ divided by $(w - x)$. Both lines have a slope of $\frac{v - z}{w - x}$. Therefore, the lines are parallel.