Question

the slope of \\(\overleftrightarrow{pq}\\) is \\((v - z)\\) divided by \\(\square\\)
the slope of \\(\overleftrightarrow{pq}\\) is \\(\square\\) divided by \\((w + a) - (x + a)\\).
both lines have a slope that is \\(\square\\) divided by \\(\square\\)
therefore, the lines are \\(\square\\) (options: parallel, perpendicular, intersecting)
(image shows points: q(w + a, v + b), q(w, v), p(x + a, z + b), p(x, z) on a coordinate plane)

Explanation:

First Blank (Slope of \(\boldsymbol{\overrightarrow{PQ}}\)):

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For points \(P(x, z)\) and \(Q(w, v)\), the change in \(y\) is \(v - z\), and the change in \(x\) is \(w - x\). So the slope of \(\overrightarrow{PQ}\) is \(\frac{v - z}{w - x}\), meaning the denominator is \(w - x\).

Second Blank (Slope of \(\boldsymbol{\overrightarrow{P'Q'}}\)):

For points \(P'(x + a, z + b)\) and \(Q'(w + a, v + b)\), the change in \(y\) is \((v + b) - (z + b)=v - z\) (the \(b\) terms cancel). The change in \(x\) is given as \((w + a) - (x + a)\), so the numerator of the slope is \(v - z\).

Third and Fourth Blanks (Slope of Both Lines):

Simplify the slope of \(\overrightarrow{P'Q'}\): \(\frac{(v + b) - (z + b)}{(w + a) - (x + a)}=\frac{v - z}{w - x}\) (the \(a\) and \(b\) terms cancel). So both lines have a slope of \(\frac{v - z}{w - x}\), meaning the numerator is \(v - z\) and the denominator is \(w - x\).

Fifth Blank (Relationship of Lines):

If two lines have the same slope, they are parallel. Since both lines have slope \(\frac{v - z}{w - x}\), they are parallel.

Filling the Blanks:

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

Final Answers (Filled Blanks):

1. \(w - x\)
2. \(v - z\)
3. \(v - z\); \(w - x\)
4. parallel

Answer:

First Blank (Slope of \(\boldsymbol{\overrightarrow{PQ}}\)):

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For points \(P(x, z)\) and \(Q(w, v)\), the change in \(y\) is \(v - z\), and the change in \(x\) is \(w - x\). So the slope of \(\overrightarrow{PQ}\) is \(\frac{v - z}{w - x}\), meaning the denominator is \(w - x\).

Second Blank (Slope of \(\boldsymbol{\overrightarrow{P'Q'}}\)):

For points \(P'(x + a, z + b)\) and \(Q'(w + a, v + b)\), the change in \(y\) is \((v + b) - (z + b)=v - z\) (the \(b\) terms cancel). The change in \(x\) is given as \((w + a) - (x + a)\), so the numerator of the slope is \(v - z\).

Third and Fourth Blanks (Slope of Both Lines):

Simplify the slope of \(\overrightarrow{P'Q'}\): \(\frac{(v + b) - (z + b)}{(w + a) - (x + a)}=\frac{v - z}{w - x}\) (the \(a\) and \(b\) terms cancel). So both lines have a slope of \(\frac{v - z}{w - x}\), meaning the numerator is \(v - z\) and the denominator is \(w - x\).

Fifth Blank (Relationship of Lines):

If two lines have the same slope, they are parallel. Since both lines have slope \(\frac{v - z}{w - x}\), they are parallel.

Filling the Blanks:

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

Final Answers (Filled Blanks):

1. \(w - x\)
2. \(v - z\)
3. \(v - z\); \(w - x\)
4. parallel