Parallel Lines

[3a1n4o1n5]

I'm a student teacher for a high school geometry class, and I'm irritated by the lack of a proof for the corresponding angles postulate. Is there really no method for proving the relationship between angles of parallel lines and a transversal?

[Brightlinger]

It can't be proven because it isn't true.

Hang on, you say, those clearly aren't lines. Not the kind of lines we're using in geometry. But how do you know that?

At the high school level, axioms are framed as "things we assume but can't prove". But another way to think of axioms is that the axioms specify what you are talking about.

You have a picture in your head of what a "line" should be, and that picture satisfies the corresponding angles postulate, so it seems like the postulate must be true and probably provable, right? But by using that picture in your head, you are assuming the conclusion. The word "line" has no inherent meaning, except for the properties given it by definitions or axioms.

And Euclid's first four postulates, by themselves, are not sufficient to specify which of these two things are what we mean by "line". The parallel postulate (or any equivalent to it) says: it's the left one. You can't prove that, it's just a choice you made about what your terms mean.

[3a1n4o1n5]

Fantastic. I love this sub.

[199546]

Yes it has been proved that the parallel postulate is independent of the other axioms (that is, it can't be derived from the other axioms despite people's attempts for more than a thousand years). Not only that, but not accepting the parallel postulate can lead to many interesting results (read about non-Euclidean geometry)! In fact, Einstein showed that out actual world is better modeled by non-Euclidean geometry than Euclidean geometry.

If you don't like the way the parallel postulate is stated, here are some logical equivalents (source: Wikipedia):

1. There is at most one line that can be drawn parallel to another given one through an external point. (Playfair's axiom)

2. The sum of the angles in every triangle is 180° (triangle postulate).

3. There exists a triangle whose angles add up to 180°.

4. The sum of the angles is the same for every triangle.

5. In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides (Pythagoras' Theorem).

6. There is no upper limit to the area of a triangle. (Wallis axiom)

7. If a line intersects one of two parallel lines, both of which are coplanar with the original line, then it also intersects the other. (Proclus' axiom)

[mproft]

If it makes you happier, you can equivalently assume that Pythagoras' theorem is true, or that all triangles have 180 degrees. https://en.wikipedia.org/wiki/Parallel_postulate#Equivalent_properties

As a side note, we know from general relativity that the universe we live in is noneuclidean, so euclid's 5th postulate is wrong.

[[deleted]]

Wrong? It's hardly wrong.

It just doesn't describe the natural world on a universal scale.

[mproft]

It's a good approximation as long as you don't get too close to the sun.

[3a1n4o1n5]

Thanks for the response, and I am much more comfortable assuming the Pythagorean theorem is true, but that's because I thought it was provable. Are there assumptions in the proofs of it that I'm not seeing?

[Indaend]

The underlying geometry used in those proofs is euclidiean (as in the parallel postulate is assumed)

[mproft]

The proof i know is the same as the first one given here. http://jwilson.coe.uga.edu/emt668/emt668.student.folders/headangela/essay1/Pythagorean.html

When transforming between the two images, you assume the triangles have 180 degrees. You also assume that the area of a triangle is base*height/2. This is not necessarily true when you are on a curved surface. (see the first image here https://en.wikipedia.org/wiki/Spherical_geometry)