Question

proving 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.

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

Step3: Compare the slopes

Since the slope of \(\overleftrightarrow{PQ}\) is \(\frac{v - z}{w - x}\) and the slope of \(\overleftrightarrow{P'Q'}\) is \(\frac{v - z}{w - x}\), both lines have a slope that is \((v - z)\) divided by \((w - x)\).

Step4: Determine the relationship between the lines

If two non - vertical lines have the same slope, they are parallel.

Answer:

The slope of \(\overleftrightarrow{PQ}\) is \((v - z)\) divided by \((w - x)\). The slope of \(\overleftrightarrow{P'Q'}\) 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.