How do we know/prove a slope of a line is constant?

[lamp23]

Before I just accepted that the slope of a line is constant, but I'm wondering if there is an even more fundamental definition of a line.

If one draws two right triangles with a certain Δx and Δy as the legs and wants to prove that the connection of the two hypotenuses is a straight line, then should one prove that the angle between them is 180°?
http://uploadpic.org/storage/2011/thumb_em0Ys5txnYiOhPNXJVSS0UDJe.jpg
I have drawn the original picture in purple and then by SAS one can prove the two triangles congruent and then prove that the corresponding angles ∅ are congruent. Once one proves there is a right angle adjacent to it and a (90-∅) adjacent to that, the sum gives 180°.

 

[Number Nine]

If $f(x) = ax + b$ then $f'(x) = a$, thus the slope is constant. No need to make it any more complicated than that.

 

[lamp23]

If $f(x) = ax + b$ then $f'(x) = a$, thus the slope is constant. No need to make it any more complicated than that.

Then you are assuming that f(x)=ax+b is the equation of the line. Yet in the derivations for an equation of a line that I have seen (one in Stewart's Calculus), that equation is derived from assuming a =Δy/Δx is constant.

 

[Mark44]

And what about the line whose equation is x = 2? This isn't even a function, let alone one that is differentiable, and yet it is a line.

 

[Deveno]

i have to ask, first, for you: what constitues an acceptable definition of a line?

there is more to the question than meets the eye, and what you will regard as an acceptable proof, depends on what you will allow as "given".

to underscore my point, in euclidean geometry, often lines are NOT defined, but are assumed to have certain properties instead (line is a "primitive concept" and any conceivable object with the properties of a line, is said to be a model for a line).

it is not hard to show, that for any set in the plane satisfying:

L = {(x,y): ax+by = c} (where a,b and c are "constants"). that the equation:

y₂ - y₁= m(x₂ - x₁)

has a unique solution m that holds for any pair (x₁,y₁), (x₂,y₂) in L; unless b = 0, in which case NO m will work.

but perhaps this is not what you're looking for, without more information, i cannot say.

 

[HallsofIvy]

Before I just accepted that the slope of a line is constant, but I'm wondering if there is an even more fundamental definition of a line.

If one draws two right triangles with a certain Δx and Δy as the legs and wants to prove that the connection of the two hypotenuses is a straight line, then should one prove that the angle between them is 180°?
http://uploadpic.org/storage/2011/thumb_em0Ys5txnYiOhPNXJVSS0UDJe.jpg
I have drawn the original picture in purple and then by SAS one can prove the two triangles congruent and then prove that the corresponding angles ∅ are congruent. Once one proves there is a right angle adjacent to it and a (90-∅) adjacent to that, the sum gives 180°.

You don't need that the two triangles are congruent- only similar. That way, you can use different length $\Delta x$ and get a different $\Delta y$. But because the hypotenuses of the two right triangles are the same line, and the two horizontal sides are parallel, by "corresponding angles" from geometry, we get that the two angles you have labeled "$\phi$" are congruent so the triangles are similar. Then the ratios of corresponding sides are the same. Since the slope is the ratio of two sides, it is the same at every point no matter what "rise" and "run" you use.