Math Is Fun Forum

Al-Allo

Member

Registered: 2012-08-23

Posts: 324

Straight lines which join the ends of equal and parallel straight line

I have a small question regarding proposition 33 of the elements of Euclid. http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI33.html

We want to prove that two lines joining equal parallels at their ends are themselves equals and parallels. The only thing I don't understand is why does Euclid use the side-angle-side theorem to prove that the remaining angles are equal to one another ? Why doesn't he use the proposition 29 (A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the sum of the interior angles on the same side equal to two right angles.) to prove that all alternate angles are equal to one another?

In my opinion, I think it's because were considering only at the begining the "finite" straight lines BA-DC, so this limits us only to two angles ABC and BCD.

Thank you

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Straight lines which join the ends of equal and parallel straight line

From what I understand, there is little difference between the two methods.  Both rely on alternate angles.  In Euclidean geometry there is often more than one way to prove something.  As long as he only uses axioms and theorems that have already been proved, what does it matter ? 

Bob

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