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I've noticed something while trying to prove the properties of parallel lines, and the properties that a triangle has 180 degrees. To prove the properties of parallel lines, such as alternate angles, you need to use the property that a triangle has 180 degrees. To prove a triangle has 180 degrees however, you need to use the properties of parallel lines. This really bothers me because of how circular it is. They are both reliant on each other to be true, and do not logically show, without being reliant on each other, why triangles have 180 degrees, and why parallel line properties are true.

So what I hoping for here is a way to prove parallel line properties without using the fact that a triangle has 180 degrees, or a way to prove triangles have 180 degrees without using parallel line properties. This way, things will be logical to me, and make sense.

Thanks in advance.

Ethan Chan
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  • In hyperbolic or spherical geometry, many parallel properties break down and triangles don't have $180^\circ$ angles. – Michael Burr Jun 04 '18 at 06:53
  • The Euclidean parallel postulate is logically equivalent to the property that angles of a triangle add to $180^\circ$. You can see these and other equivalent properties here. – Blue Jun 04 '18 at 06:54
  • Possibly you're asking for a proof that one of these properties is satisfied by the standard plane $\mathbb R^2$? – Chris Culter Jun 04 '18 at 07:11
  • “Properties of parallel lines” encompasses a lot of facts. Are you sure that the same property that was required by the proof of the triangle sum is one of the properties that is proved by using the triangle sum? In which book did you find this being done? – David K Jun 18 '18 at 17:53

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This interesting question touches on a serious question about the axioms for Euclid's geometry. The parallel lines axiom (often called the "parallel postulate") seems to have a flavor different from the others. For centuries mathematicians tried to prove it. Along the way they discovered many theorems that are equivalent to it - you could use any one of them as an axiom instead of the parallel postulate and end up with the same geometry. Among those theorems:

  • The angles of a triangle sum to two right angles.

  • There is a pair of similar triangles that are not congruent.

  • Two lines parallel to the same line are parallel to each other.

  • The Pythagorean theorem.

The wikipedia page linked above lists more. I recommend it.

Eventually Lobachevski and Gauss and Bolyai and others discovered that you could do nice geometry even when the parallel postulate failed - thus discovering (or inventing) non-Euclidean geometry.

Edit in answer to a comment.

You ask

Did non-Euclidean geometry prove or disprove the postulate?

The answer is "neither". What the invention of non-Euclidean geometry proved is that it is impossible to prove or disprove the parallel postulate starting from the other axioms. More formally: if it's possible to reach a contradiction from the other axioms along with the negation of the parallel postulate then that contradiction can be reached from the other axioms and the parallel postulate.

The Greeks found the parallel postulate pretty clearly "true" in the "real world", so they built it into their abstraction of that world - the Euclidean plane. But it did bother mathematicians from then on, hence the attempts to prove it and the eventual proof that you can't.

We don't in fact know if the parallel postulate is true in the space we live in. Einstein's theory of general relativity says it's not when matter is present. But even where matter is relatively rare, space may be curved in a sense that mathematicians have made precise. If it is curved then you have to look at a pretty large volume to tell - much as you have to look at a large area of the surface of the earth to detect that you are not on a Euclidean plane. (The Greeks did know that.) Search for is our world Euclidean to read more.

https://www.google.com/search?q=is+our+world+euclidean

Ethan Bolker
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To prove parallel line properties you only need the parallel lines axiom, stating that through a given point there is a UNIQUE line parallel to a given line. The existence of such a line can be proved via Euclid's exterior angle theorem: if a line forms congruent alternate angles with another line, then those lines are parallel.

The converse of that theorem can then be proved by RAA: if lines $a$ and $b$ are parallel, suppose by contradiction they do not form congruent alternate interior angles with a transversal at $A$ and $B$. Then, as explained above, you could construct another line $b'$ through $B$ parallel to $a$ and that would violate the unicity axiom.

Intelligenti pauca
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