Final answer:
Elastic cartilage contains elastic fibers, allowing it to maintain shape and offer flexibility, which is exemplified by the cartilage in the external ear.
Step-by-step explanation:
The type of cartilage that maintains structure and shape while permitting extensive flexibility is elastic cartilage. This tissue contains a large amount of elastic fibers in addition to collagen and proteoglycans, which gives it the ability to provide firm support with great elasticity. An example of elastic cartilage in the body is the cartilage found in the external ear, which allows it to bend and then return to its original shape.