Congruent figures have equal areas. Why?

Question

When we are first introduced to the idea of congruent polygons (generally triangles) in school, we often define congruent figures as "figures which have the same shape and size". However, today, congruence seems to be (generally) described formally by isometries. How can we prove that congruent figures have equal areas using the (modern) definition of congruence? Also, I don't think that Euclid ever proves that if we "cut" a polygon into any number of parts, the sum of the areas of the parts is equal to the area of the original. I'm not sure whether it even would have mattered to Euclid, since I have heard that he did not distinguish between polygons and their areas, and the case for polygons would have been covered by "the whole is equal to the sum of the parts. Is it possible to prove this? Or do we take it as another axiom?