Question

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

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 \(P(x,z)\) 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 \(m=\frac{(v + b)-(z + b)}{(w + a)-(x + a)}=\frac{v - z}{w - x}\)

Step3: Compare the slopes

Since the slope of \(\overleftrightarrow{PQ}\) is \(\frac{v - z}{w - x}\) and the slope of \(\overleftrightarrow{P'Q'}\) is also \(\frac{v - z}{w - x}\), when two lines have the same slope, they are parallel.

Answer:

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