Question

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

Explanation:

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

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For points \(P(x,z)\) and \(Q(w,v)\), \(m_{PQ}=\frac{v - z}{w - x}\)

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

For points \(P'(x + a,z + b)\) and \(Q'(w + a,v + b)\), \(m_{P'Q'}=\frac{(v + b)-(z + b)}{(w + a)-(x + a)}=\frac{v - z}{w - x}\)

Answer:

Both lines have a slope that is \((v - z)\) divided by \((w - x)\). Since the slopes of \(\overleftrightarrow{PQ}\) and \(\overleftrightarrow{P'Q'}\) are equal (\(m_{PQ}=m_{P'Q'}=\frac{v - z}{w - x}\)), the lines are parallel.