Talk:Transversal (geometry)

Suggestions for page

My suggestions for this page following this discussion.

I think it is important to define a transversal properly (as it is done in this page) and then state (something like)

In Euclidean geometry, Euclid’s parallel postulate guarentees that two lines are parallel only if the interior non-adjacent angles on the same side of any transversal are supplementary, that is, they sum to 180°.

(This is just a simple restatement of the first theorem below.)

Then two images of transversals through non-parallel and parallel lines with non-supplementary and supplementary angles denoted.

Then we could define the angle pairs of transversals: complementary, alternating formally and give separate images for each type of pair as in: Parallel lines – Transversals (in MK). I can do this if nobody objects.

BTW: In MK we have a term for “interior non-adjacent angles on the same side of any transversal”. Is there such a term in English?

Then, we could have the sections on Theorems and e.g. Menelaus' theorem, ... . Lfahlberg (talk) 06:52, 9 September 2013 (UTC)[reply]

I have no objections to expanding the article as you suggest as long as we don't get into WP:NOTTEXTBOOK territory. To me it's unfortunate that this rather arcane terminology has found its way into high school geometry syllabi (see, e.g. www.mt-jfk.com/syllabus/geometrysyllabus.doc‎). It's not useful in real life and modern mathematics doesn't really use the term. It's historically significant and interesting if you're the kind of person who likes reading TL Heath's translation of Euclid, but otherwise not so much. So I don't think we need to give examples and diagrams for every definition. --RDBury (talk) 22:51, 9 September 2013 (UTC)[reply]

I don't know how to make the images a bit smaller and keep the frames... Anyone can feel free to do this. Lfahlberg (talk) 16:14, 16 September 2013 (UTC)[reply]

Critique/question on page

I'm quite new to editing in Wikipedia. So please excuse me if I don't follow all the protocols. Neither am I a mathematician even if I do have a B.D. in math. So maybe my question is considered trivial by the connoiseurs. However I'm a bit bothered by the fact that the definition of alternate angle does not agree with the definition in Euclid Elementa, Def. 1.46, see [1]. When I tried to check the german definition it seems to be in line with Euclid, so at least there seems to be different definitions in the German vs the English version, provided of course that I haven't misunderstood the German page. [2]. If the original Euclidean definition is correctly referenced, it seems a bit pointless to start deviating from a definition which has been unchanged the last 2300 years.

In the original definition it's by definition only the internal angle pairs that are considered to be alternate angles. The corresponding opposite angles of course have the same measure. However that fact doesn't make them alternate by definition, in the same manner that an opposite angle isnt identical to the angle it's opposing. Anorell (talk) 22:31, 10 February 2018 (UTC)[reply]

A couple of comments are in order. First of all, while Euclid uses the term "alternate angles" he does not define it. There are only 23 definitions in book I of the Elements and this term is not there. Your source for what is contained in the Elements is not a literal translation and many things are cast into modern terms. In his commentaries, Proclus claims that what Euclid meant by the term was a pair of angles on opposite sides of the transversal and opposite sides of the two lines the transversal crosses. This is precisely what the German page says and is in complete agreement with our page. In the pivotal Proposition I.27, Euclid only assumes that a transversal makes equal alternate interior angles, which is all that is needed to prove the result. He could have started by assuming equal alternate exterior angles, but by considering vertical angles he would reduce it to the other case, so this would not be needed. The second point to make is that it really doesn't matter how Euclid defined things, except to put things in an historical context. Definitions change over time, usually for good reasons like reducing ambiguity and becoming more precise. Mathematics grows and from time to time needs to shed itself of outmoded concepts. --Bill Cherowitzo (talk) 20:57, 11 February 2018 (UTC)[reply]