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Is the point order of the triangles really relevant when it comes to congruence?

Let us assume that: $\triangle ABC \cong \triangle DEF$

Which means that there is a congruence correspondence $ABC \longleftrightarrow DEF$. Intuitively I'd say that $\triangle CBA \cong \triangle DEF$ as well because we're dealing with the same triangles.

I'm currently working myself through Geometry by Harold R. Jacobs. In it the definition of congruence is:

Two angles are congruent iff there is a correspondence between their vertices such that the corresponding sides and corresponding angles of the triangles are equal.

In the book the author states that if $\triangle GHI \cong \triangle LKJ$, then there is a correspondence $GHI \longleftrightarrow LKJ$, but there is no correspondence $GHI \longleftrightarrow JKL$ therefore we cannot write $\triangle GHI \cong \triangle JKL$. ($\triangle GHI$ and $\triangle LKJ$ are neither isosceles, nor equilateral in this specific example).

Is this really the case?

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    $\begingroup$ Well, this is true, because when you write that $\triangle ABC \cong \triangle DEF$, you imply that the angle $BAC$, for instance, is the same as $EDF$. This is additional information from congruence, this is the way to point out which angles are congruent in the triangles. $\endgroup$ – Martigan Nov 24 '14 at 15:52
  • $\begingroup$ I see! Now I see the reasoning behind this. Thank you very much! $\endgroup$ – geomquestion Nov 24 '14 at 15:54

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