What particular textbook and edition are we discussing? I’d be interested in how the concept is presented and if it is a matter of definition (e.g. this is what similarity means) vs notation (e.g. for clarity/full credit/ease of grading when expressing that two triangles are similar, you should always order the corresponding vertices in the same order.)
It’s notation. In the book, ABC[tilde]DEF means A = D, B = E, C = F. So if A doesn’t equal D, then ABC[slash-tilde]DEF, even if the two triangles are similar.
I’ll do a couple scans later today.
Why are you shy about sharing the actual title and edition information of this particular book you keep referring to, which several people who have responded to you have already asked about?
Not all, but at least a statistically significant subset.
Sure that’s true, but it would mean a statistically significant subset for each state - states do have different textbook standards from other states.
And states like California or Texas, given their size, have major sway among textbook publishers. And a range in sizes for each district within any state. There’s a bit of work in determining what a statistically significant subset would be. It’s not exactly straightforward.
Either way, there’s nothing like a single source website that would have something esoteric like that listed on a webpage, and it’s a non-trivial exercise to deduce (even with generous error bars) a reasonable estimate.
Yes, a photo of the actual page and text would be helpful so we can interpret it ourselves.
On page 137 it says it’s customary. then on the middle of p141 it seems to be mandatory
Link doesn’t work for me. Tell us the full name of the text book, the author, and the page number. Why are you so coy about this?
It works for me. Here’s the cover image.
The exact quote:
Then, on the other page scanned,
I guess the question is, “If someone has written ‘△ABC ~ △DEF,’ can you necessarily conclude that ∠A ≅ ∠D?” And the answer is “Yes you can, if you know that they are following the convention that ‘corresponding vertices are listed in the same order.’”
I’m totally confused. We must be talking past each other.
If one triangle has two angles the same as (any) two angles in another triangle then the third is the same too. That’s how triangles work.
If the three angles of a triangle have the same angles as the three angles of another triangle, then they are similar -whether they are inverted, rotated, or mirror image of each other or all of the above.
And similar means the edges of one are not necessarily the same length as corresponding edges of the other.
If the USA is teaching something different, then that is truly something different from what I learned 60 years ago in Canada. It all comes down to the definition of “similar”.
Maybe that’s the first thing this thread should establish, that we are talking about similar “similar”.
There cannot be any question about what “similar” means, but there may be a question of whether all the students grok what a similarity is after completing a particular course of instruction.
I think the complaint is that if ABC is similar to DEF then the textbook implies it is not similar to a renamed triangle DFE (or any other permutation) because B does not equal F and C does not equal F, when it is, in reality, similar. I do not see the textbook implying any such thing
I mean, so far as I can parse the OP.
Agree w @pulykamell just above. and @Thudlow_Boink 4 posts up.
There is no controversy here beyond a willful (?) desire to overconstrue one less than perfect sentence in some particular textbook.
It is difficult to write prose about mathematics precisely. Hence all the fancy notation to achieve the desired unambiguity. It’s equally a mistake to try to read that prose with greater precision than was put into writing it.
Sorry–it was like 3 a.m. when I wrote this. This should be B does not equal F; and C does not equal E.
You can have a concept of similarity without correspondence, or you can have a concept of similarity with correspondence. But similarity without correspondence is a useless concept, because there’s nothing you can do without the correspondence.
+1 vote.
Having now seen the relevant part of the textbook, this seems more a misinterpretation by the OP than any sort of systemic feature of the US educational system.
There’s no implication – the textbook says plainly that ABC[tilde]DEF only if A = D, B = E and C = F. So if ABC[tilde]DEF is true, ABC[tilde]EFD is false, according to the book.
ABC is a 3-4-5 triangle. DEF is a 6-8-10 triangle. Is ABC[tilde]DEF?
What’s the answer to that question, according to the correct interpretation? Can’t be yes, since we don’t know whether A = D. Can’t be no. So the correct interpretation is, we don’t know.
It’s a misinterpretation because the book does not explicitly state it is a general principle.
It provides a specific example (the figure on the page) and makes the correct statement that angle A corresponds to D, B to E, C to F. That is absolutely true for the triangle on the page.
It does not explicitly state the order the angles are listed matters for any other case, i.e. making that extrapolation is an interpretation and not a correct one.