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I need help with a terminology definition. If we say "R is a reflection that exchanges the sides a and b in some triangle", does it mean sides a and b have the same length and the reflection maps one on the other exactly? Or does it mean something else, like swapping places (where the lengths are not necessarily equal? Though that would make the statement trivially true hence redundant...)

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The latter isn't even well-defined. I'd go with the former. –  anon Feb 9 '12 at 21:33
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I'd go with your first option, that the sides are the same length and the image of each of those sides is equal to the preimage of the other side.

It might be easier to tell for sure given more context for the statement.

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