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I was wondering whether there is a proof of SSS Congruence Theorem (and also whether there is one for SAS and ASA Congruence Theorem). In my textbook, they are treated as a postulate, or one that we just accept as truth without basis. Is the 3 theorems for similar triangles really just postulates, or are there any proof of them? I tried searching online but I couldn't find one.

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  • $\begingroup$ Related (Duplicate?): "Proof for SSS Congruence?". My answer there has a link to a useful page. $\endgroup$ – Blue Nov 19 '16 at 6:49
  • $\begingroup$ If congruence means that angles and sides must be the same, then something like the cosine rule, which relates all three sides to one specific angle, should immediately show why $SSS$ rule implies congruence. Similar use of sine/cosine rule should also explain why the other rules work (for example, if SAS is given, then the third side can be found by cosine rule, and you have the SSS case, for example)/ $\endgroup$ – астон вілла олоф мэллбэрг Nov 19 '16 at 6:51
  • $\begingroup$ @астонвіллаолофмэллбэрг I am of the opinion that trigonometric formulae like cosine rule are generally taught after teaching basic geometric properties like congruence and similarity because to prove pythagoras one needs similarity. $\endgroup$ – vidyarthi Nov 19 '16 at 6:56
  • $\begingroup$ @астонвіллаолофмэллбэрг: Taking the Law of Cosines as a prerequisite for these results is pretty extreme. SAS and SSS are true even in so-called "neutral geometry" ---the common ancestor of both Euclidean and Non-Euclidean geometry--- and are established (one assumed, one proven) before Law of Cosines makes an appearance. (I'll note that Euclidean and Non-Euclidean geometries have very distinct Laws of Cosines.) $\endgroup$ – Blue Nov 19 '16 at 6:59
  • $\begingroup$ @Blue I agree with both of you, but if we did not consider it an axiom, then how would another model proof proceed? That's why I wrote about the cosine rule (in Euclidean geometry), because I felt it was somewhat defeatist to say "it is an axiom". It is, as the previous answer below says, and I accept that now. $\endgroup$ – астон вілла олоф мэллбэрг Nov 19 '16 at 7:21
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The SAS criterion for congruence is generally taken as an axiom. From this, and using other postulates of Euclid, we can derive the ASA and SSS criterion. The proof proceeds generally by contariction.

For ASA criterion, we cut one of the sides so as to make it equal to corresponding part of the other triangle, and then derive contradiction.

Similarly, for SSS criterion, we arrive at contradiction by cutting one of the angles and making it equal ti the corresponding angle of the other triangle.

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