I am starting to learn geometrical proofs, and I have come across the Symmetry property of segment congruence (if $AB$ is congruent to $CD$, then $CD$ is congruent to $AB$).

One of the exercises in my book tell me to prove this using the property of reflexivity (segment AB is congruent to segment $AB$), and the theorem that is if segment $AB$ is congruent to segment $CD$ and segment $AB$ is congruent to segment $EF$, then segment $CD$ is congruent to segment $EF$.

I am really stuck and have no idea how to prove this. Please help or give me a starting point.

Thank you.


1 Answer 1


Here's a more a complete answer. There are two givens, the Euclidean and reflexive properties of congruence.

i. $ \text{If}\space AB\cong CD \space\text{and}\space AB\cong EF, \space\text{then} CD\cong EF. \space\space\space\space\text{(Euclidean property)}$

ii. $ AB \cong AB.\space\space\space\space\text{(reflexive property)}$

Now, the statement you have prove is the symmetric property: $ \text{if}\space AB\cong CD, \space\text{then} \space CD \cong AB. $ To prove it, you start by assuming $AB\cong CD$. Remember you are also given $AB \cong AB$. But if both $AB\cong CD \space\text{and}\space AB\cong AB$, then by the Euclidean property $CD \cong AB$. QED.

  • $\begingroup$ I know I have to use the reflexive property, but I don't know what to do with it :( $\endgroup$
    – Paul Filch
    Sep 25, 2013 at 5:47
  • 1
    $\begingroup$ This is going to sound condescending at first, but I promise I'm actually about to teach you a profoundly simple lesson in how to think like a mathematician. You know you have to use a particular statement in some way, but you don't know to do with it. Now, what are you supposed to do with the statement? The only thing you can do: you state it. So, can you state the reflexive property? $\endgroup$
    – David H
    Sep 25, 2013 at 5:58
  • $\begingroup$ I solved it :) Thank you $\endgroup$
    – Paul Filch
    Sep 25, 2013 at 6:58

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