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."