Final answer:
The genioglossus muscle is commonly responsible for the relaxation of the tongue and epiglottis during anesthesia. This muscle's relaxation can cause airway obstruction, which is why endotracheal intubation is used to maintain an open airway during surgical procedures.
Step-by-step explanation:
The muscle that commonly causes the tongue and epiglottis to relax during the induction of anesthesia is the genioglossus muscle. This muscle originates on the mandible and allows the tongue to move downward and forward. Under anesthesia, the drugs used relax a majority of the body's muscles, including the genioglossus, which can lead to the relaxation of the tongue and epiglottis, potentially obstructing the airway. To control this and avoid complications such as aspiration, medical professionals utilize a procedure called endotracheal intubation to maintain an open airway to the lungs during surgery.