Question

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

Explanation:

Step1: Calculate the slope of \(\overleftrightarrow{PQ}\)

The formula for the slope \(m\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For points \(P(x,v)\) and \(Q(w,z)\), the slope of \(\overleftrightarrow{PQ}\) is \(\frac{z - v}{w - x}\). So the slope of \(\overleftrightarrow{PQ}\) is \((z - v)\) divided by \((w - x)\).

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 \(y_2=v + b\), \(y_1=z + b\), \(x_2=w + a\), \(x_1=x + a\). Then \(m=\frac{(v + b)-(z + b)}{(w + a)-(x + a)}=\frac{v - z}{w - x}\). So the slope of \(\overleftrightarrow{P'Q'}\) is \((v - z)\) divided by \((w + a)-(x + a)\)

Step3: Compare the slopes

Since the slope of \(\overleftrightarrow{PQ}\) is \(\frac{z - v}{w - x}\) and the slope of \(\overleftrightarrow{P'Q'}\) is \(\frac{v - z}{w - x}\) (equivalent to \(\frac{-(z - v)}{w - x}\), but if we consider the general form of the ratio for slope calculation, both slopes have the form \((z - v)\) (or \((v - z)\) which is a scalar - multiple in the sense of ratio) divided by \((w - x)\). When two lines have the same slope, they are parallel.

Answer:

1. \((w - x)\)
2. \((v - z)\)
3. \((v - z)\) (or \((z - v)\) depending on the sign, but as a ratio \(\frac{z - v}{w - x}\) and \(\frac{v - z}{w - x}\) are equivalent in terms of the condition for parallelism as \(\frac{z - v}{w - x}=-\frac{v - z}{w - x}\), and the non - zero denominator \((w - x)\) makes the ratio (slope) relationship hold for parallelism)