Proving triangle sum theorem and parallel lines and corresponding angles

Question

To prove triangle sum theorem, you either have to accept that corresponding angles created by a transversal through two parallel lines are equal or you have to prove that, but when proving that they are equal then you have to accept that the triangle sum is 180 degrees or prove it but that takes me back to where I started from. So my question is, how can we consider these proofs to be proofs if they are codependent. When proving parallel and corresponding theorem, I assumed that two corresponding angles where equal, but the lines weren't parallel, but this requires you to accept that a triangle has 180 degrees angle sum for when two lines aren't parallel and intersected by a transversal then they form a triangle. Are my "proofs" too basic, or am I missing something? For when I try to prove one of them, I have to just accept that the other is a fact, without proving, since you cannot prove any of these without accepting that the other is a fact. Please let me know if you require more information from me. But I just want to know how can I accept these as proofs when I have to accept the other as a fact without proving it.

Answer 1

==== short answer ====

Euclids 5th postulate is an axiom. It can not be proven. i) The sum of the angles of a triangle ii) the tranversal of parallel lines being equal iii) exactly one parallel line through a point not on the line. Are three equivalent statements. Any one can be an axiom to prove the other two. But one must be assumed axiomatically.

===== long answer =====

You aren't "missing" anything. Do you understand the concept of axiomatic, logical systems?

You can't prove/know/define anything except what you prove or define from simpler concept. Thus at the bottom you need bedrock assumed axioms that are taken as given. Ex: lines are made up of points (what is a "point"?). You can construct a line between any two points (why?). Lines in a plane are either parallel or intersect at one point. etc.

We can't avoid having fundamental axioms that we must take as a given but we can work of paring our axioms down to an essential few with no redundancy. For example: In algebra we might think we have three basic fundamentals: 1) $a(b+c) = ab + ac$, 2) $0*a = 0$, and all numbers, 3) $a$ have an inverse $-a$ so that $a+(-a) = 0$.

It turns out those 3) are redundant and we only need 1) and 3) and we can prove 2). (Pf: $0*a = (0+0)a = 0*a + 0*a$ so $0 = 0*a +(-0*a) = (0*a+0*a) +(-0*a) = 0*a$). So we have two fundamental axioms 1) and 3) and a proven result 2).

Now we could just as easily had 1) and 2) be the axioms and 3) be the proven result. (... actually, in this example we'd need a few other axioms about multiplication, but I hope the idea is clear, namely...) We can have a set of some axioms and some basic proven results for a consistent logic system. Or we could have a different equivalent system were some of the basic proven results are treated as axioms, and instead the assumed axioms are proven.

So Euclidean Geometry is an axiomatic system and one of the axioms is Euclid's 5th Postulate.

Euclis's 5th Postulate basically says if sum the interior angles of two lines is less than a linear angle (two right angles) the lines will intersect on the side where the sum is less than a linear angle (i.e. add to 180).

That is an axiom. As immediately consequence we have that if lines don't meet then the sum of the interior angles will be a linear angle.

From that we can have two alternative but equivalent definitions of parallel i) two lines are parallel if they never meet ii) two lines are parallel if their interior angles are linear (add to 180).

And from that axiom we can prove a slew of things including

i) Given a line and one point off the line there is precisely one parallel line to the point.

ii) the sum of the angles of a triangle is a linear (add to 180).

Now take these three statements:

1. if sum the interior angles of two lines is less than a linear angle (180) the lines will intersect on the side where the sum is less than a linear angle (180 degrees).
2. Given a line and one point off the line there is precisely one parallel line to the point.
3. the sum of the angles of a triangle is a linear (add to 180).

If you assume any one of these, you can prove the other two.

Thus the three statements are equivalent.

But none of them can be proven by themselves.

Or, I should say, given the other axioms of geometry which do not address the issue of parallelism at all (instead the address things about lines existing betweeen two points, that circles of any radius uniquely exist at a center, etc). We need an axiom about parallelism if we are going to talk about parallelism.

So what should be our basic axiom? Well, any of those three equivalent ones will do. Euclid, for reasons we can only speculate (but IMO is intuitive and makes sense) choose 1).

So yes, you are correct. You can only prove one if you assume the other. This would be circular if one didn't have an axiom to fall back upon. But we do.

BTW. If we tossed out Euclids 5th postulate, and even if we replace it with axioms that say the exact opposite, the remaining system has been proven to be consistent (but with very different results). Thus we have Non-Euclidean Geometry.

Answer 2

Equality of angles follows almost immediately from the original Euclidean version of the parallel postulate: Suppose the corresponding angles are unequal. It follows (given a little calculation) that the sum of the interior angles on one side of the transversal is less than two right angles; hence (by Euclid's version of the postulate) the two lines intersect on that side of the transversal, contradicting the assumption of parallelism. The only 'angle sum' property required is that opposite-side angles always sum to two right angles.