Proof #2: Two Distinct Parallel Lines Have the Same Slope (Proof by Contradiction)

proof goal

Prove that two distinct parallel lines have the same slope

proof method

Proof by contradiction

proof assumption

Assume that two distinct parallel lines do not have the same slope

line labeling

Let the two lines be l and n

parallel condition

Lines l and n are said to be parallel

slope assumption

Suppose the lines have different slopes, so m₁ ≠ m₂

line l equation

y

line n equation

y

intersection test

To find if the lines intersect, set their equations equal: m₁x + b

solving for x

Subtract m₂x from both sides → (m₁ − m₂)x

isolating x

x

x-meaning

Since m₁ ≠ m₂, there is a valid real x value

interpretation of x

A valid x means a point of intersection exists

contradiction discovered

Parallel lines cannot intersect, so a point of intersection contradicts their parallelism

logical result

The assumption that parallel lines can have different slopes is false

final conclusion

Two distinct parallel lines must have the same slope