Some? many? most? students in the US nowadays are taught that if triangle ABC is similar to triangle DEF then angle A equals angle D and angle B equals angle E. In other words, if scalene triangle ABC is similar to triangle DEF, then it’s not similar to triangles DFE or EDF etc.
As I recall, decades ago we learned that if two triangles were similar, then they were similar. Didn’t matter what you called them. Are most students in the US, and the rest of the world, now taught the new way?
Two questions, with what I guess are the answers now deemed correct:
Triangle ABC has sides 3, 4, and 5; triangle DEF has sides 6, 8 and 10. Are the two triangles similar? (Yes)
Is triangle ABC similar to triangle DEF? (We don’t know – probably not.)
It is hard to answer this without an excerpt from the presumably awful textbook which does not make clear what similar figures are. Two given triangles are either “similar” or not “similar”, and presumably “triangle DEF” and “triangle DFE” are two names for the very same triangle.
I have seen textbooks illustrating just what it means for, say, pentagon ABCDE to be similar to FGHIJ, namely, that there should be a specific similarity mapping one to the other and therefore a specific correspondence of consecutive vertices and sides in order.
From where is that quoted? Similar triangles (or other shapes) are not required to be congruent: you are free to translate, scale, rotate, and reflect them. In particular, it is the ratio between the lengths of the parts that is preserved, not the lengths themselves. Euclid simply states (for rectilinear figures) that the corresponding angles are equal and the respective sides in proportion.
Come to think of it, we could talk about two circles being similar, or two line segments. However, that is not very useful, and it is no surprise textbooks are going to talk about similar triangles.
Triangles are similar regardless of how they’re labeled. But in order to actually do anything with similar triangles, you need the correspondence, and it’s very difficult to establish the correspondence without corresponding labeling. In other words, if you say that “Triangle ABC is similar to triangle DEF”, well, FDE is the same triangle as DEF, but the statement is interpreted as meaning “ABC is similar to DEF, with A corresponding to D, B corresponding to E, and C corresponding to F”.
The gist of my question is, how are students taught these days, in the US and elsewhere.
No typo in my original post. According to a Common Core textbook, triangle ABC is similar to triangle DEF only if angle A equals angle D and angle B equals angle E. If all we know is triangle ABC has sides 3, 4 and 5 and DEF has sides 6, 8 and 10, then we don’t know whether angle A equals angle D – I purposely left that out of the question. So we don’t know ABC is similar to DEF, according to Common Core. All we know is ABC is similar to one of the six permutations of D and E and F, and not similar to the other five.
Incidentally, the textbook I’m looking at is cunningly designed to put students to sleep. Must have been designed by teachers who like a quiet classroom.
Maybe I’m missing something, but all 3 corresponding sides are in proportion, so according to the SSS Theorem, they would indeed be similar. You can then use the Law of Cosines to find the actual angles. When I taught this I established the proportions first for each pair of corresponding sides, then (for the high schoolers) I would introduce both laws (+ Law of Sines) so they could solve the entire triangle. Annnnd I LOVED getting into the SSA cases, because they tended to be a trip-I’d have them construct the triangle out of the 3 known parts, then notice that the third side can just flap like a barn door in the breeze because we don’t know what its angles are and it may not even each the other side at all.
Presumably those are the angles that are in correspondence, i.e. the measure of angle A = measure of angle D, measure of angle B = measure of angle E, and measure of angle C = measure of angle F. And that the order matters.
But clarification from the OP would be safest.
My $0.02 is that the triangles are similar. However you choose to name them is up to you. It doesn’t matter if you call it triangle ABC or triangle Bob.
If you want to make the order you list the vertices relevant, it’s on you to make that distinction clear, because that won’t be universally acknowledged. Unambiguous language is important, after all. Is there some text or some teacher making that non-universal distinction? Maybe. But that doesn’t mean all children are taught that way.
My question is, are students nowadays taught that if scalene triangle ABC is similar to triangle DEF, then triangle ABC is not similar to triangle FDE etc?
In any case, you are allowed to say “the two triangles are similar” because in that sentence you haven’t spelled out the order of the vertices – right? So even tho the two triangles are similar, ABC is not similar to FDE. That’s the way Common Core works.
We can go even simpler. Is the triangle ABC congruent to triangle ACB? In one sense, you can say “Well of course it is; it’s the same triangle”. But in another sense, that’s the basis of the simplest proofs of the two Isosceles Triangle Theorems.
Common Core has standards, but it doesn’t dive into that level of detail.
For example, it lists triangle similarity as something students should understand for geometry, but it does not mandate “how” subjects should be taught, mainly being concerned with “what”, with examples.
A Common Core compliant textbook can (and apparently does) teach this particular concept in a way that may not be universally accepted.
You can look up the standards. The ones for triangle similarity are on page 77 on this pdf: