Slope Criterion for Parallel Lines

    • Prove the slope criterion for parallel lines

The slope criterion for parallel lines states

    • Two non-vertical lines are parallel if and only if they have the same slope

Let's look at some proofs

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Proof 1

Here's my favorite. This proof just uses algebra

The slope intercept form of a line is y = mx + b.

    • m is the slope

    • b is the y-intercept

Subtracting both sides by b then dividing both sides by x, we could rewrite this equation as

We have two lines here. We'll assume they same slope, then see what happens.

However, the red line has a y-intercept at A, and the blue line has a y-intercept at C. Using our modified equation and substituting these y-intercepts in for b, we can write the equations for the two lines.

Now, remember what do to if you want to find out where two lines intersect? Solve for a variable in one equation, then substitute it into the other equation. We could do that here, but since both equations are already solved for m, it's easier to just set them equal to each other.

=

Now lets simplify. Multiply both sides by x.

y - A = y - C

Subtract y from both sides.

- A = - C

Multiply both sides by -1.

A = C

And we get that the lines intersect when A = C...

...Wait a second. how can A ever equal C? Only if the lines are right on top of each other, and that wouldn't really be parallel lines, that would be two lines that are the exact same line. So A never equals C. Why did we set these equations equal to each other? To see where they intersect. And our answer was impossible.

Since all we said about these two lines is that their slopes are the same, that means lines with the same slope never intersect. And what are lines that never intersect called? Parallel lines. Putting it all together, lines with the same slope are parallel.

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Proof 2

This proof is a pure geometry proof, with no algebra. We start with two lines, m and n, and we're told they are parallel. We need to prove that they have the same slope. Remember the definition of slope

First, we'll set up the drawing.

  • Pick any two points on line m and connect them at point E to form a right triangle.

  • Extend line AE so that it crosses line n at point c.

  • Pick a point F on extended line AE so that CF is congruent to AE

  • Connect F and D

Now what do we know?

  • ∠AED ≅ ∠CFD (all right angles are congruent)

  • AE ≅ CF (we drew it that way)

  • ∠BAE ≅ ∠DCF (corresponding angles)

    • Since lines m and n are parallel, line AE is a transversal, which lets us use the corresponding angles theorem.

Well, this gives us enough to use ASA for triangle congruency. Then using CPCTC, we know that DF = BE and CF = AE.

Now, let's write the slopes of the lines

slope of line m =

slope of line n =

But since we just said that DF = BE and CF = AE, these slopes are the same. That means that parallel lines have the same slope. Q.E.D.