Question

the slope of $overleftrightarrow{pq}$ is $(v - z)$ divided by $(w - x)$ $checkmark$
the slope of $overleftrightarrow{pq}$ is $\boldsymbol{\text{dropdown}}$ divided by $(w + a) - (x + a)$.
both lines have a s $\boldsymbol{(v + b) - (z + b)}$ $checkmark$ divided by $(w - x)$ $checkmark$
therefore, the lines are parallel $checkmark$
image: coordinate plane with points $p(x, z)$, $p(x+a, z+b)$, $q(w, v)$, $q(w+a, v+b)$; $x$-axis (horizontal), $y$-axis (vertical); two lines ($overleftrightarrow{pq}$ and $overleftrightarrow{pq}$); “hint” button at bottom right

Explanation:

Step1: Recall Slope Formula

The slope of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is \(\frac{y_2 - y_1}{x_2 - x_1}\). For \(\overrightarrow{P'Q'}\), the endpoints are \(P'(x + a, z + b)\) and \(Q'(w + a, v + b)\).

Step2: Calculate Numerator for Slope of \(\overrightarrow{P'Q'}\)

Using the slope formula, the numerator (change in \(y\)-coordinates) is \((v + b) - (z + b)\). Simplifying this: \(v + b - z - b = v - z\).

Step3: Calculate Denominator for Slope of \(\overrightarrow{P'Q'}\)

The denominator (change in \(x\)-coordinates) is \((w + a) - (x + a)\). Simplifying this: \(w + a - x - a = w - x\).

Step4: Compare Slopes

The slope of \(\overrightarrow{PQ}\) is \(\frac{v - z}{w - x}\). The slope of \(\overrightarrow{P'Q'}\) is \(\frac{(v + b) - (z + b)}{(w + a) - (x + a)}=\frac{v - z}{w - x}\), which is the same as the slope of \(\overrightarrow{PQ}\). When two lines have the same slope, they are parallel.

Answer:

The slope of \(\overrightarrow{P'Q'}\) is \(\boldsymbol{(v + b) - (z + b)}\) divided by \((w + a) - (x + a)\). Both lines have a slope of \(\frac{v - z}{w - x}\), so the lines are \(\boldsymbol{\text{parallel}}\).