Talk:Congruence (geometry)

message: An easy way to remember that SSA (Side-Side-Angle) does not prove congruence for triangles is to simply reverse the order of the letters. --Jediknil
all of it is so wrong  Unsigned comment added by 86.1.74.239 at 17:34 on January 21st, 2013‎
I mean Angle-Side-Side (No I will not say the acronym) is technically a legitimate name for SSA  — Preceding unsigned comment added by Elitematterman (talk • contribs) 13:53, 17 April 2019 (UTC)
thread_title:  A handy mnemonic 

message: "In less formal language, two sets are congruent if they have the same shape and size, but are in different positions (for instance one may be rotated, flipped, or simply moved)."
Should we specify two different sets? Every set is congruent to itself.--Syd Henderson 01:16, 22 September 2006 (UTC)
replies: text: Yes, that seems a sound policy.  Do you think we need an article on dilation, or is it adequately covered elsewhere?  Dbfirs 23:10, 9 December 2009 (UTC)
replies: text: I just found out there is a good link to dilation on the similarity page. I see no reason to have one here. Not really applicable to congruence in my opinion. JackOL31 (talk) 05:33, 10 December 2009 (UTC)
replies: text: Ah, so we do have an article!  I'm happy to leave congruence as it is without a link.  Clearly congruence is a special case of similarity by the definition given for dilation.  Dbfirs 08:38, 10 December 2009 (UTC)
thread_title:  Equality 

message: Dbfirs, I see you undid and edit for AAS. I presume your position is that AAS stands on its own and needs no further explanation. However, I don't see how the addl explanation negatively affects the article. In the US, AAS is typically taught by first noting the two corresponding angle congruencies (Given), the third angle must be congruent (No-Choice Thm), followed by the congruent corresponding sides (Given), thereby showing the triangles are congruent (ASA). Perhaps presented as additional information rather than justification is a good compromise. JackOL31 (talk) 02:39, 13 November 2009 (UTC)
replies: text: Yes, ASA is not taught in the UK (except as a special case of AAS).  We could add a comment to that effect if you wish.  Dbfirs 10:56, 13 November 2009 (UTC)
replies: text: Please bear in mind that what is or is not taught in the UK shouldn't necessarily come into play. Whether the information is accurate (or at least perceived accurate) and whether it is relevant to the article is the concern. Yes, I believe adding the ASA connection would be appropriate. You've seen the Anon editor's words and my words, what did you have in mind? JackOL31 (talk) 22:08, 14 November 2009 (UTC)

text: I'll go ahead and make the change after the holidays. JackOL31 (talk) 00:40, 18 November 2009 (UTC)
replies: text: My proposed addition to the AAS postulate. Additionally, if two pairs of angles of two triangles are equal in measurement, then the remaining pair of angles are also equal in measurement (all angle measurements in a triangle sum to 180). With all three pairs of angles equal in measurement, ASA now applies. JackOL31 (talk) 18:05, 21 November 2009 (UTC)
replies: text: Would it not also be logical to say that ASA is a special case of AAS?  I consider that it reduces the clarity of AAS to add a "proof" in terms of a special case.  Dbfirs 18:56, 23 November 2009 (UTC)

text: Ah! I see.  In the UK "AA(corr)S" includes both AAS and ASA.  I assume that they are separate in the USA because ASA is taught as a separate theorem.  Do you teach AAS in the USA?  I hope you are having a happy Thanksgiving (something else that we don't have in the UK).  Dbfirs 09:31, 26 November 2009 (UTC)
replies: text: Hmmm, I have always seen the acronyms portrayed as indicating relative positioning. That is, one starts at the first triangle component and moves in a consistent direction to the next adjacent component. Therefore, AAS (and its mirror SAA) results in the first and last components being opposite. Accordingly, the same holds true for SSA and its mirror (assume "and its mirror" from now on). Given the different treatments for SSA and SAS in the article, I would make the same distinction for AAS and ASA. I can't speak for all US school districts and/or math teachers, but I believe the general approach is to introduce the SSS, SAS and ASA postulates. Then reintroduce the idea that having two angles for a triangle means we have the third angle. Therefore, given two triangles with two pairs of corresponding congruent angles, the remaining pair of corresponding angles are also congruent (No-Choice Theorem). Now, if we are given that a pair of corresponding sides are also congruent, we can apply the ASA postulate and state that AAS also determines congruency. Subsequently, I guess one could now note that two pairs of congruent angles and any pair of corresponding congruent sides (which I will abbreviate as A-A-S) indicates congruency between two triangles. In other words, I would hold (IMHO) that the A-A-S grouping is more of an artificial grouping rather than a postulate in and of itself. I don't see AAS as a special case of A-A-S since both AAS and ASA are "special cases" and require independent checking. They just happen to render the same congruency results allowing for the artificial grouping. For S-S-A, we know this not to be the case since SSA and SAS do not render the same congruency results. To answer your question directly, I have never seen A-A-S stated as some type of postulate (eh, before now). Anyway, that's my take on it. Re: Thanksgiving Day holidays - I did have a good time, thank you. I visited my son at the university he is attending - 3,470 km away. JackOL31 (talk) 00:47, 2 December 2009 (UTC)
replies: text: I can see the logic and rigour of the US approach.  AA(corr)S was taught just as a sufficient condition for congruence in the UK, without considering order, and without dividing it into a theorem/postulate and a corollary.  Perhaps we should keep the US approach with just a note on the UK sufficient condition, since most students who are looking for the formal approach will be from the USA.  There is no university in the UK that could be even half that distance from any residence! At least your travel costs are much lower than they are here!   Dbfirs 13:55, 4 December 2009 (UTC)

text: It was taught as AA (corr)S two angles and a corresponding side) in the UK (in fact I'd never seen ASA and AAS until I read Wikipedia), but I'll do a bit of research to see how widespread and current this mnemonic is now.  Since Euclidean geometry is not a major part of the modern curriculum in the UK, it seems sensible to retain the American mnemonics in the Wikipedia articles.  Perhaps we could add just a note at the end, rather than confuse the list of mnemonics known in the USA.  I'm happy with your parenthetical, though I wonder if AAS should be relegated to a note, or is AAS taught as a separate condition in the USA?  Dbfirs 17:11, 5 December 2009 (UTC)
replies: text: Now I don't feel so bad about trapeziums and rhomboids. I can say that my kids learned AAS as a separate topic, however they used the same textbook. I recall learning it as SAA, but not the details of where in relation to the other topics. Based the original editor's input that started this discussion, that person appears to have learned AAS after ASA. I think it's fine to leave it there. JackOL31 (talk) 19:14, 5 December 2009 (UTC)
thread_title:  Determining congruence 

message: From my Mathematics textbook, R.H.S. (Right angle, hypoteneuse, side) can also determine congruent triangle. Am I right?--Sap00acm (talk) 09:20, 26 November 2008 (UTC)
replies: text: That's correct. This is described in the third paragraph of the "Angle-Side-Side" section. -- Jitse Niesen (talk) 10:28, 26 November 2008 (UTC)
thread_title:  R.H.S? 

message: I deleted the statement that the study of invariant properties is the essence of geometry. Geometry can be studied without being concerned with unchanging properties. Therefore, the study of invariant properties is not the essence of geometry.Lestrade (talk) 18:34, 15 April 2009 (UTC)
thread_title: Essence

message: :Note that congruences alter some properties, such as location and orientation, but leave others unchanged, like distance and angles.
Distance is variant. Presumably it means length. Unless I am mistaken.  —Preceding unsigned comment added by George Makepeace (talk • contribs) 13:08, 5 November 2010 (UTC)
thread_title:  Distance is invariant 

message: The article begins by stating that two objects are congruent if either object can be repositioned so as to coincide precisely with the other object. Then it adds that this is equivalent to an an isometry, i.e., a combination of translations, rotations and reflections. This is not the same thing, you cannot reposition in space a right hand glove in order to coincide with a left hand one, a reflection is needed. Csoliverez, February 10, 2012, 21:02 UTC.  — Preceding unsigned comment added by Csoliverez (talk • contribs) 21:03, 10 February 2012 (UTC)
replies: text: It's not inconsistent. In geometry, repositioning two objects so that they coincide means repositioning by means of a rigid transformation (isometry), which includes reflections.--fwchapman (talk) 02:31, 8 October 2013 (UTC)
thread_title:  Inconsistency in the definition 

message: I have issues with the following sentence: "However, in spherical geometry and hyperbolic geometry (where the sum of the angles of a triangle varies with size) AAA is sufficient for congruence on a given curvature of surface." I'm no expert, but I see 2 problems. First, AAA would prove similarity, but not congruence. Second, I think the sum of the angles of a triangle varies, not with size, but in some other way. I'm not sure what it is. But here's an easy example. On a spherical model of Earth, draw 2 triangles. Each triangle has a vertex at a pole, say 30 degrees; also each has 2 right angles. Sides: 2 sides of each triangle are on longitudes, 1 is on a latitude. Whatever the latitude is, AAA applies; the triangles are similar, the sum of angles is the same (210 degrees), but the sizes (areas, side lengths) vary. Thus, the quoted sentence isn't  correct as it stands. Would someone who knows spherical and hyperbolic geometry please fix this part of the article? Oaklandguy (talk) 06:19, 17 May 2012 (UTC)
replies: text: The statement that AAA proves congruence in spherical geometry is correct. In spherical geometry, all similar triangles on spheres of the same radius are actually congruent. Your counterexample doesn't work because the lines in spherical geometry are great circles (circles whose centers coincide with the center of the sphere). That means the only latitude that can be the side of a triangle is the equator. The two sides from a pole to the equator both have length one quarter of the circumference of the sphere. The triangles in your example are congruent by SAS, which is valid in spherical geometry.--fwchapman (talk) 02:26, 8 October 2013 (UTC)
thread_title:  AAA (Angle-Angle-Angle) in spherical and hyperbolic geometry 

message: I have moved this material here from Spherical trigonometry without attempting to integrate it correctly. Further edits may help.  Peter Mercator (talk) 13:53, 28 July 2013 (UTC)
thread_title:  Spherical triangles 

message: I think the given example of a family of spherical triangles with side lengths of π, π/2, and π/2 actually results in degenerate triangles. The two sides through the point p on the equator actually lie on the same great circle. Thus, each triangle in the family actually has only two non-collinear sides, which is a degenerate triangle. We need three non-collinear sides to create a non-degenerate triangle. Is this correct? If so, can anyone suggest how to remedy this?
--fwchapman (talk) 20:48, 7 October 2013 (UTC)
replies: text: Thanks. Done. Duoduoduo (talk) 15:34, 11 November 2013 (UTC)
thread_title:  Congruent triangles on a sphere 

message: Hiya! While I think the section on congruence of triangles in the plane is extremely important, I think perhaps we might first focus on congruence of figures in the plane in general.
I think we should add that an isometry is a combination of rigid motions. The actual article on isometry jumps right into metric spaces.
... by an isometry, i.e., a combination of rigid motions, namely translations, rotations and reflections.
BTW: I don't see the need for plurals in this last, but it doesn't matter. 
I think we should add (here is a reference ], although probably the given definition of congruence suffices): 
*two line segments are congruent if they are the same length.
*two angles are congruent if they are have the same measure (regardless of direction).
*two circles are congruent if they have the same diameter. 
*two plane figures are congruent if their corresponding sides are congruent, their corresponding interior angles are congruent, their corresponding diagonals are congruent, their perimeters are equal and their areas are equal. 
BTW: I wasn't exactly sure how to word this last statement, but being mathematically absolutely correct down to the last detail defeats the purpose ...
I think we should add (something like):
*two distinct plane figures are congruent if we can cut them out of paper and then match them up completely. Turning the paper over is permitted.
I think we should add a section: Checking congruence of plane figures
*First, match and label the corresponding vertices of the two figures.
*Second, draw a vector from one of the vertices of the one of the figures to the corresponding vertex of the other figure. Translate the first figure by this vector so that these two vertices match.
*Third, rotate the translated figure about the matched vertex until one pair of corresponding sides matches.
*Fourth, reflect the rotated figure about this matched side until the figures match. 
If at anytime the step cannot be completed, the figures are not congruent.
I might add the images at right. Please feel free to improve the text below them. I have a no-callout gif uploaded for multi-language translation of the animated gif; the 2nd image is a png - anybody can feel free to make an svg (I balked at the grid). The colors seem to be good for standard colorblindness.
Thanks for reading. I would appreciate any comments to these suggestions.
Lfahlberg (talk) 14:56, 16 October 2013 (UTC)
replies: text: Since no one has commented, I plan to make these changes.
Lfahlberg (talk) 09:13, 25 October 2013 (UTC)
replies: text: My only reservation would be the concept of congruent angles.  Angles of equal measure are congruent in a Hilbertian sense as long as we delimit the angles using infinite rays, but representations of angles of equal measure need not be congruent in the normal sense that you explain so clearly.  This doesn't matter in the USA where everyone is taught Hilbertian angles, but could cause confusion in the UK where angles of equal measure are normally just called "equal", not congruent.  Dbfirs 21:26, 25 October 2013 (UTC)
replies: text:  Hiya all. We discussed "congruent angles" with Dbfirs and agreed on the format in the changed article. BTW: I changes the svg at the top to be 4 triangles as I was concerned that a reader might get the false impression that the 4th figure was not congruent just because it was not a triangle. I desperately want to credit MADe (the original author of the old svg: https://en.wikipedia.org/wiki/File:Congruentie.svg) on this svg, but every time I tried to upload my svg to wiki commons with the credit it would not accept it so I finally gave up. If anyone knows how to add the credit to my image file, please do (or tell me how). Thanks. Lfahlberg (talk) 18:41, 27 October 2013 (UTC)
thread_title:  Some comments, suggestions, images 

message: In the section "Determining congruence of polygons", the sentence beginning " Third, ..." contains the phrase " corresponding sides". Please link this to corresponding sides.
208.50.124.65 (talk) 18:30, 20 June 2014 (UTC)
replies: text: Done with thanks, NiciVampireHeart 19:08, 20 June 2014 (UTC)
thread_title:  Semi-protected edit request on 20 June 2014 

message: Multiple high school Geometry textbooks (Dr. Ron Larson's textbook)used in the USA refer to RHS as HLItalic text (Hypotenuse-Leg). consider including this as this could avoid confusing high school students seeking homework help via Wikipedia.
Sincerely,
High School Teacher (13 years)
208.67.117.129 (talk) 19:29, 17 October 2014 (UTC)
replies: text: Done Stickee (talk) 06:23, 19 October 2014 (UTC)
thread_title:  Semi-protected edit request on 17 October 2014 

message: Hello fellow Wikipedians,
I have just added archive links to  on Congruence (geometry). Please take a moment to review my edit. If necessary, add  after the link to keep me from modifying it. Alternatively, you can add  to keep me off the page altogether. I made the following changes:
*Added archive https://web.archive.org/20131111162553/http://146.163.152.131/teaching/projects/creech_final.pdf to http://146.163.152.131/teaching/projects/creech_final.pdf
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Cheers. —cyberbot IITalk to my owner:Online 02:26, 28 August 2015 (UTC)
thread_title:  External links modified 

message: 97.93.99.144 (talk) 23:18, 7 December 2016 (UTC)
replies: text: Not done: No request made. -- Dane talk  04:30, 8 December 2016 (UTC)
thread_title:  Semi-protected edit request on 7 December 2016 

message: "In the School Mathematics Study Group system SAS is taken as one (#15) of 22 postulates." in Determining Congruence Seems out of place considering its placed in ASA instead of SAS. I can make the change myself but I want someone to either confirm or put my suspicions to rest  NikkeKatski [Elite] (talk) 16:16, 17 April 2019 (UTC)
replies: text: I can see why you think this is out of place, but the solution to that problem is not as simple as moving it to a different heading. The entire discussion of which of these are treated as theorems and which as postulates needs to be pulled out from the ASA section and handled after all the acronyms have been defined. I will try to do this soon. --Bill Cherowitzo  (talk) 18:25, 18 April 2019 (UTC)
replies: text: Thank you c: --NikkeKatski [Elite] (talk) 17:41, 19 April 2019 (UTC)
thread_title:  Im no expert but...