The left and right common carotid aa. supply blood to the head and neck. Their courses are the same, with the exception that the right common carotid is a branch of the brachiocephalic a. (which is the first major branch of the arch of the aorta), while the left common carotid branches directly from the arch of the aorta. The common carotid aa. branch to form the internal and external carotid aa.