Evolutionary theory predicts that related organisms will share similarities that are derived from common ancestors. Similar characteristics due to relatedness are known as homologies. Homologies can be revealed by comparing the anatomies of different living things, looking at cellular similarities and differences, studying embryological development, and studying vestigial structures within individual organisms.
In the following photos of plants, the leaves are quite different from the "normal" leaves we envision.
Each leaf has a very different shape and function, yet all are homologous structures, derived from a common ancestral form. The pitcher plant and Venus' flytrap use leaves to trap and digest insects. The bright red leaves of the poinsettia look like flower petals. The cactus leaves are modified into small spines which reduce water loss and can protect the cactus from herbivory.
Another example of homology is the forelimb of tetrapods (vertebrates with legs).
Frogs, birds, rabbits and lizards all have different forelimbs, reflecting their different lifestyles. But those different forelimbs all share the same set of bones - the humerus, the radius, and the ulna. These are the same bones seen in fossils of the extinct transitional animal, Eusthenopteron, which demonstrates their common ancestry.
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Pitcher plant image courtesy of Mr. C. Lee, Malesiana Tropicals; Poinsettia image courtesy of USDA; Venus' flytrap image courtesy of R. Sacilotto, www.pitcherplant.com; Barrel cactus image courtesy of Matthew K. Gray.
Understanding Evolution © 2016 by The University of California Museum of Paleontology, Berkeley, and the Regents of the University of California