I believe they do, a congruent figure have to be the same shape and size, so they would have to have equal surfaceareas.
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The Definition of Congruent Figures (which is a proof) says that if two figures have corresponding sides congruent and corresponding angles congruent, then the figures are to be congruent.
congruent
Congruent means two figures have exactly the same shape and size. If the shape is identical, but not the size , two figures are similar.
congruent
CONGRUENT