Question

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

Explanation:

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

The formula for the slope \(m\) of 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 points \(P(x,z)\) and \(Q(w,v)\), the slope of \(\overrightarrow{PQ}\) is \(\frac{v - z}{w - x}\).

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

For points \(P'(x + a,z + b)\) and \(Q'(w + a,v + b)\), the slope of \(\overrightarrow{P'Q'}\) is \(\frac{(v + b)-(z + b)}{(w + a)-(x + a)}\). Simplifying the numerator \((v + b)-(z + b)=v - z\).

Step3: Compare the slopes

Since the slope of \(\overrightarrow{PQ}=\frac{v - z}{w - x}\) and the slope of \(\overrightarrow{P'Q'}=\frac{v - z}{w - x}\) (after simplifying \(\frac{(v + b)-(z + b)}{(w + a)-(x + a)}\)), and if two lines have the same slope, they are parallel.

Answer:

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