A 61-year-old man was hit by a cricket bat in the mid-humeral region of his left arm. Physical examination reveals normal elbow motion; however, he could not extend his wrist or his metacarpophalangeal joints and he reported a loss of sensation on a small area of skin on the dorsum of the hand proximal to the first two digits. Radiographic examination reveals a hairline fracture of the shaft of the humerus just distal to its midpoint. Which of the following nerves is most likely injured?