How did palaeontologists discover the age of the Tyrannosaurus rex named Sue?
Dinosaur bones contain rings, similar to a tree, that can be counted to determine age. SUE was 28 when she died, making her the most geriatric Tyrannosaurus yet found. Her bones show signs of wear and even disease, including arthritis.
Paleontologists can make micron slices of bone and view the rings through polarized light, even on a microscope at 5-10x, similar to geologist mineral ID's.
https://mistralmtn.blogspot.fr/2013/03/bone-growth-rings.html

Wiki: Growth lines may be deposited in synchrony with endogenous biorhythms. For example, captive crocodilians exposed to constant temperature, diet, and photoperiod, still exhibit the periodic and cyclical skeletal growth banding of their wild counterparts.[4] Consequently, it is assumed by many paleontologists that the growth lines of dinosaurs reflect annual rhythms, and that they may be used to determine individual ages. However, in the large and long bones of many dinosaurian taxa, resorption of internal and external bone proceeds even as new cortical bone continues to be deposited, so that growth lines deposited early in development may need to be inferred.
http://tbrnewsmedia.com/wp-content/uploads/2015/05/Dinosaur-Growth-Rate-w.jpg