Final answer:
The bone extension that supports teeth is the alveolar process of the maxilla and mandible, secured by the periodontal ligament within the gomphosis joint.
Step-by-step explanation:
The extension of bone that supports the teeth in their functional positions is known as the alveolar process. This bony structure can be found in both the maxilla, which forms the upper jaw, and the mandible, known as the lower jaw. The alveolar process contains sockets called alveoli, where the teeth are anchored. These sockets house the roots of the teeth that are held in place by the periodontal ligament, a type of connective tissue. Additionally, the joint where the tooth is anchored into the bony socket is known as gomphosis, which is a fibrous joint. The immobility of the gomphosis renders it a synarthrosis joint, essential for the stability of the teeth.