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An "antagonist" is a muscle that acts opposite to the specific movement generated by the agonist and is responsible for controlling the motion, slowing it down and returning a limb to its initial position.
Muscles can only exert a pulling force and cannot push themselves back into their original positions. Therefore performing "return motions" may need a pair of muscles with opposite effect. As one muscle contracts, the other relaxes. This pair is called an "antagonistic pair". An example of exception is Sphincter ani externus muscle. An example of Antagonisic pairs is the Bicep and Tricep as to contract your tricep lengthens(relaxes) while the Bicep contracts to lift your arm.
"Reverse motions" need antagonistic pairs located in opposite sides of a joint or bone, including abductor-adductor pairs and flexor-extensor pairs. These consist of an extensor muscle, which "opens" the joint (i.e., increasing the angle between the two bones) and a flexor muscle, which does the opposite to an extensor muscle.
Antagonism is not an intrinsic property. It is a role, played depending on the motion. If the motion is reversed, agonist and antagonist swich roles. A flexor muscle is always flexor. But in flexion, it is always agonist and in extension, it is always antagonist. An extensor muscle is agonist in extension and antagonist in flexion.
When the biceps are contracting, the triceps relax, and stretch back to their original position. The opposite happens when the triceps contract.