A rigid motion is a transformation that preserves length and angle measures. Therefore, if we transform a figure A to figure A' then, by the definition of rigid motion, every part of figure A is congruent to corresponding parts of figure A'. If we do another rigid motion to transform A' to A'', then again by the definition of rigid motion, every part of A' is congruent to its corresponding part of A''. So by the Transitive Property of Congruence, every part of figure A is congruent to corresponding to every part of figure A''.
