Question

the slope of (overrightarrow{pq}) is ((v - z)) divided by (square). the slope of (overrightarrow{pq}) is (square) divided by ((w + a) - (x + a)). both lines have a slope that is (square) divided by (square). therefore, the lines are (square). image: coordinate plane with points ( q(w+a, v+b) ), ( q(w,v) ), ( p(x+a, z+b) ), ( p(x,z) ), ( x )- and ( y )-axes, and a “hint” button

Explanation:

Step1: Find the denominator for slope of $\overrightarrow{PQ}$

The slope formula is $\frac{y_2 - y_1}{x_2 - x_1}$. For points $P(x,z)$ and $Q(w,v)$, the change in $x$ is $w - x$. So the denominator for the slope of $\overrightarrow{PQ}$ (which has numerator $v - z$) is $w - x$.

Step2: Find the numerator for slope of $\overrightarrow{P'Q'}$

For points $P'(x + a,z + b)$ and $Q'(w + a,v + b)$, the change in $y$ is $(v + b)-(z + b)=v - z$. So the numerator for the slope of $\overrightarrow{P'Q'}$ is $v - z$.

Step3: Simplify the denominator for slope of $\overrightarrow{P'Q'}$

Simplify $(w + a)-(x + a)$: $w + a - x - a = w - x$. So both slopes have numerator $v - z$ and denominator $w - x$.

Step4: Determine the relationship between the lines

If two lines have the same slope, they are parallel. Since both lines have slope $\frac{v - z}{w - x}$, they are parallel.

Answer:

1. The slope of $\overrightarrow{PQ}$ is $(v - z)$ divided by $\boldsymbol{w - x}$.
2. The slope of $\overrightarrow{P'Q'}$ is $\boldsymbol{v - z}$ divided by $(w + a)-(x + a)$.
3. Both lines have a slope that is $\boldsymbol{v - z}$ divided by $\boldsymbol{w - x}$.
4. Therefore, the lines are $\boldsymbol{parallel}$.