Question

the slope of \overleftrightarrow{pq} is (v - z) divided by (z - v).
the slope of \overleftrightarrow{pq} is (v + b)-(z + b) divided by (w + a)-(x + a).
both lines have a slope that is (v - z) divided by (w - x).
therefore, the lines are parallel.

Explanation:

Step1: Recall the 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 the line \(\overleftrightarrow{PQ}\) with points \(P(w,v)\) and \(Q(x,z)\), the slope \(m_{PQ}=\frac{v - z}{w - x}\).

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

For the line \(\overleftrightarrow{P'Q'}\) with points \(P'(w + a,v + b)\) and \(Q'(x + a,z + b)\), use the slope formula \(m_{P'Q'}=\frac{(v + b)-(z + b)}{(w + a)-(x + a)}\). Simplify the numerator \((v + b)-(z + b)=v - z\) and the denominator \((w + a)-(x + a)=w - x\).

Answer:

Since the slope of \(\overleftrightarrow{PQ}\) is \(\frac{v - z}{w - x}\) and the slope of \(\overleftrightarrow{P'Q'}\) is also \(\frac{v - z}{w - x}\) (after simplifying \(\frac{(v + b)-(z + b)}{(w + a)-(x + a)}\)), two lines with the same slope are parallel.