The BBC says that the biceps and triceps are antagonistic muscle pairs. Muscles can pull, but they cannot push. If there was only one muscle, then the limb wouldn't be able to get back into its original position. Antogonist muscle pairs are necessary for proper movement.
The forearm is controlled by the biceps and triceps contractions. The BBC explains that the forearm moves downs when the triceps contract, and it moves up when the biceps contract. The movements succeed when the muscles get shorter.
Another example of an antagonistic muscle pair is the quadriceps and hamstrings. The BBC says that these muscles are responsible for extending and contracting the knee. The quadriceps contract to pull the lower leg forward, and the hamstrings contract to pull the lower leg backwards. The motion allows the leg to swing back and forth. The pair allows the arm to lift objects and put them down.
According to Aworkouts.com, the back and chest muscles are antagonistic muscle pairs. The back helps the body perform pull-ups, while the chest allows the body to do push-ups. The lower back and the abdominal muscles act as antagonistic muscle pairs as well. The abs contract forward when bending over, and the lower back contracts to pull the body upright.