Question

Determine the missing information in the paragraph proof.

Given: Line PQ contains points (w, v) and (x, z) and line P'Q' contains points (w + a, v + b) and (x + a, z + b). Lines PQ and P'Q' are parallel.

Prove: Parallel lines have the same slope.

On a coordinate plane, 2 lines are shown. Line Q P goes through (w, v) and (x, z). Line Q prime P prime goes through (w + a, v + b) and (x + a, z + b).

Since slope is calculated using the formula m = StartFraction v 2 minus v 1 Over x 2 minus x 1 EndFraction, the slope of both lines is equivalent to ________. It is given that the lines are parallel, and we calculated that the slopes are the same. Therefore, parallel lines have the same slopes.



StartFraction v minus z + b Over x minus z + a EndFraction
StartFraction w minus x + a Over v minus z + b EndFraction

Answer 1

To complete the proof, we need to find the correct expression for the slopes of the given lines PQ and P'Q'.

### Step 1: Calculate the slope of Line PQ
The points on line PQ are (w, v) and (x, z).

Using the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \):

For line PQ, we take:
- \( (x_1, y_1) = (w, v) \)
- \( (x_2, y_2) = (x, z) \)

The slope \( m_{PQ} \) is given by:
\[
m_{PQ} = \frac{z - v}{x - w}
\]

### Step 2: Calculate the slope of Line P'Q'
The points on line P'Q' are (w + a, v + b) and (x + a, z + b).

For line P'Q', we take:
- \( (x_1, y_1) = (w + a, v + b) \)
- \( (x_2, y_2) = (x + a, z + b) \)

The slope \( m_{P'Q'} \) is given by:
\[
m_{P'Q'} = \frac{(z + b) - (v + b)}{(x + a) - (w + a)} = \frac{z - v}{x - w}
\]

### Conclusion
Both slopes of lines PQ and P'Q' are equal:
\[
m_{PQ} = m_{P'Q'} = \frac{z - v}{x - w}
\]

Thus, we conclude the proof.

### Final Missing Information
The expression that should fill the blank in the paragraph proof is:
\[
\frac{z - v}{x - w}
\]

Thus, the completed proof will read:
"Since slope is calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), the slope of both lines is equivalent to \(\frac{z - v}{x - w}\). It is given that the lines are parallel, and we calculated that the slopes are the same. Therefore, parallel lines have the same slopes."