Question

the slope criteria
the slope of pq is (v - z) divided by (z - v)
the slope of pq is divided by (w + a) - (x + a).
(v + b) - (z + b)
(z + b) - (v + b)
(v + b) + (z + b)
both lines have a divided by
therefore, the lines are

Explanation:

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

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 \( P(w,z) \) and \( Q(v,v) \), the slope of \( \overleftrightarrow{PQ} \) is \( \frac{v - z}{w - v} \).

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 \( y_2=(v + b) \), \( y_1=(z + b) \), \( x_2=(w + a) \), \( x_1=(x + a) \). So the slope is \( \frac{(v + b)-(z + b)}{(w + a)-(x + a)}=\frac{v - z}{w - x} \). Since \( PQ \) and \( P'Q' \) are parallel (because they have the same slope \( \frac{v - z}{w - v} \) (for \( PQ \)) and \( \frac{v - z}{w - v} \) (after simplifying \( \frac{(v + b)-(z + b)}{(w + a)-(x + a)}=\frac{v - z}{w - v} \) assuming \( x = v \) and \( w=w \))).

Answer:

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