# Similar triangles: how do US students get taught about them? **URL:** \ **Category:** Factual Questions\ **Created:** [July 23, 2025, 10:15pm UTC](https://boards.straightdope.com/t/similar-triangles-how-do-us-students-get-taught-about-them/1021241 "2025-07-23T22:15:23Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![Timz](https://avatars.discourse-cdn.com/v4/letter/t/6a8cbe/32.png) [@Timz](https://boards.straightdope.com/u/Timz)\ **Post date:** [July 23, 2025, 10:15pm UTC](https://boards.straightdope.com/t/similar-triangles-how-do-us-students-get-taught-about-them/1021241/1 "2025-07-23T22:15:23Z")
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.) ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [July 24, 2025, 3:37am UTC](https://boards.straightdope.com/t/similar-triangles-how-do-us-students-get-taught-about-them/1021241/2 "2025-07-24T03:37:43Z")
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. ---
**Author:** ![jnglmassiv](https://avatars.discourse-cdn.com/v4/letter/j/f07891/32.png) [@jnglmassiv](https://boards.straightdope.com/u/jnglmassiv)\ **Post date:** [July 24, 2025, 4:44am UTC](https://boards.straightdope.com/t/similar-triangles-how-do-us-students-get-taught-about-them/1021241/3 "2025-07-24T04:44:25Z")
CPCTC Congruent parts of congruent triangles are congruent. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [July 24, 2025, 4:58am UTC](https://boards.straightdope.com/t/similar-triangles-how-do-us-students-get-taught-about-them/1021241/4 "2025-07-24T04:58:29Z")
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. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [July 24, 2025, 7:17am UTC](https://boards.straightdope.com/t/similar-triangles-how-do-us-students-get-taught-about-them/1021241/5 "2025-07-24T07:17:27Z")
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. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 24, 2025, 1:04pm UTC](https://boards.straightdope.com/t/similar-triangles-how-do-us-students-get-taught-about-them/1021241/6 "2025-07-24T13:04:13Z")
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”. ---
**Author:** ![jnglmassiv](https://avatars.discourse-cdn.com/v4/letter/j/f07891/32.png) [@jnglmassiv](https://boards.straightdope.com/u/jnglmassiv)\ **Post date:** [July 24, 2025, 2:09pm UTC](https://boards.straightdope.com/t/similar-triangles-how-do-us-students-get-taught-about-them/1021241/7 "2025-07-24T14:09:38Z")
> [@DPRK](#): > > From where is that quoted? High school geometry, early 1990s. ---
**Author:** ![John\_DiFool](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/john_difool/32/19543_2.png) [@John\_DiFool](https://boards.straightdope.com/u/John_DiFool)\ **Post date:** [July 24, 2025, 2:25pm UTC](https://boards.straightdope.com/t/similar-triangles-how-do-us-students-get-taught-about-them/1021241/8 "2025-07-24T14:25:33Z")
> [@Timz](#): > > 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.) [my bold] Is this a typo in the 2nd case, where you meant to put “congruent”? > [@jnglmassiv](#): > > CPCTC > Congruent parts of congruent triangles are congruent. CPSTP, Corresponding parts of similar triangles are in proportion. Tho while I drilled that _concept_ into my student’s heads, it doesn’t quite have the memorable ring of the other acronym to be helpful. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [July 24, 2025, 2:36pm UTC](https://boards.straightdope.com/t/similar-triangles-how-do-us-students-get-taught-about-them/1021241/9 "2025-07-24T14:36:19Z")
> [@John\_DiFool](#): > > > [@Timz](#): > > > > Is triangle ABC **similar** to triangle DEF? (We don’t know – probably not.) > > [my bold] > > Is this a typo in the 2nd case, where you meant to put “congruent”? I _suspect_ it’s a typo where the OP meant to say “Is triangle ABC **similar** to triangle DFE?” That is, is it correct to write “△ABC ~ △DFE” ? And if that is not correct, is it because △DFE is considered to be a different triangle than △DEF? Or because similarity is not, strictly speaking, a relation between triangles but a relation between corresponding parts of triangles? Or because the symbol ~ doesn’t just mean “is similar to” but rather “is similar to, with the corresponding vertices in the order indicated”? Is that the gist of your question, @Timz? ---
**Author:** ![Timz](https://avatars.discourse-cdn.com/v4/letter/t/6a8cbe/32.png) [@Timz](https://boards.straightdope.com/u/Timz)\ **Post date:** [July 24, 2025, 4:46pm UTC](https://boards.straightdope.com/t/similar-triangles-how-do-us-students-get-taught-about-them/1021241/10 "2025-07-24T16:46:22Z")
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. ---
**Author:** ![John\_DiFool](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/john_difool/32/19543_2.png) [@John\_DiFool](https://boards.straightdope.com/u/John_DiFool)\ **Post date:** [July 24, 2025, 5:17pm UTC](https://boards.straightdope.com/t/similar-triangles-how-do-us-students-get-taught-about-them/1021241/11 "2025-07-24T17:17:25Z")
> [@Timz](#): > > 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. 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. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [July 24, 2025, 5:58pm UTC](https://boards.straightdope.com/t/similar-triangles-how-do-us-students-get-taught-about-them/1021241/12 "2025-07-24T17:58:24Z")
> [@Timz](#): > > No typo in my original post. Then I don’t understand the difference between the two questions > [@Timz](#): > > Are the two triangles similar? (Yes) and > [@Timz](#): > > Is triangle ABC similar to triangle DEF? (We don’t know – probably not.) I thought “the two triangles” _were_ ABC and DEF. Can you clarify? ---
**Author:** ![pulykamell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pulykamell/32/3166_2.png) [@pulykamell](https://boards.straightdope.com/u/pulykamell)\ **Post date:** [July 24, 2025, 6:29pm UTC](https://boards.straightdope.com/t/similar-triangles-how-do-us-students-get-taught-about-them/1021241/13 "2025-07-24T18:29:33Z")
I couldn’t understand the OP when this thread had zero replies, and I’m no closer to understanding it. Glad I’m not the only one. ---
**Author:** ![Great\_Antibob](https://avatars.discourse-cdn.com/v4/letter/g/e47c2d/32.png) [@Great\_Antibob](https://boards.straightdope.com/u/Great_Antibob)\ **Post date:** [July 24, 2025, 6:31pm UTC](https://boards.straightdope.com/t/similar-triangles-how-do-us-students-get-taught-about-them/1021241/14 "2025-07-24T18:31:51Z")
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. ---
**Author:** ![Timz](https://avatars.discourse-cdn.com/v4/letter/t/6a8cbe/32.png) [@Timz](https://boards.straightdope.com/u/Timz)\ **Post date:** [July 24, 2025, 8:17pm UTC](https://boards.straightdope.com/t/similar-triangles-how-do-us-students-get-taught-about-them/1021241/15 "2025-07-24T20:17:16Z")
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. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 24, 2025, 8:23pm UTC](https://boards.straightdope.com/t/similar-triangles-how-do-us-students-get-taught-about-them/1021241/16 "2025-07-24T20:23:28Z")
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. ---
**Author:** ![pulykamell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pulykamell/32/3166_2.png) [@pulykamell](https://boards.straightdope.com/u/pulykamell)\ **Post date:** [July 24, 2025, 8:51pm UTC](https://boards.straightdope.com/t/similar-triangles-how-do-us-students-get-taught-about-them/1021241/17 "2025-07-24T20:51:14Z")
> [@Timz](#): > > That’s the way Common Core works. Common Core does not dictate any type of pedagogy. Teachers are free to teach and set curriculum as they wish, so far as I understand it. ---
**Author:** ![Timz](https://avatars.discourse-cdn.com/v4/letter/t/6a8cbe/32.png) [@Timz](https://boards.straightdope.com/u/Timz)\ **Post date:** [July 24, 2025, 10:56pm UTC](https://boards.straightdope.com/t/similar-triangles-how-do-us-students-get-taught-about-them/1021241/18 "2025-07-24T22:56:50Z")
Okay – I should have said that’s what this textbook says, and the book says “Common Core” on it. If Common Core has no rules, what does it have in common instead? ---
**Author:** ![Great\_Antibob](https://avatars.discourse-cdn.com/v4/letter/g/e47c2d/32.png) [@Great\_Antibob](https://boards.straightdope.com/u/Great_Antibob)\ **Post date:** [July 24, 2025, 11:14pm UTC](https://boards.straightdope.com/t/similar-triangles-how-do-us-students-get-taught-about-them/1021241/19 "2025-07-24T23:14:11Z")
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: > **[Math\_Standards1.pdf](https://corestandards.org/wp-content/uploads/2023/09/Math_Standards1.pdf)** > > 1212.97 KB ---
**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\ **Post date:** [July 24, 2025, 11:52pm UTC](https://boards.straightdope.com/t/similar-triangles-how-do-us-students-get-taught-about-them/1021241/20 "2025-07-24T23:52:00Z")
Sounds like we may have identified the real issue here. Somebody wants to rant about Common Core. It’s certainly a debatable topic, perhaps even a rantworthy topic. But this category isn’t for rants. [Next page](https://boards.straightdope.com/t/similar-triangles-how-do-us-students-get-taught-about-them/1021241.md?page=2)