Antagonistic muscles do not perform their actions the way that the definition of “antagonistic” says. Rather, they simply counteract the effects of the other muscle in a particular pair.
An example of an antagonistic muscle pair is the bicep and the tricep. The bicep is responsible for contracting in such a way that the forearm is pulled back towards the body. The tricep counteracts this motion by contracting and forcing the arm back into a straight position.
The antagonistic muscle pair does not act hostile towards the other pair member, like hormones are when in an antagonistic pair, but work together in a cooperative fashion, allowing us to hold many different positions with our arms, and giving us better strength and flexibility in our arms.
Another, but similar, pair is in the leg. The hamstring and the quadricep are together an antagonistic pair. When the leg is contracted, the hamstring is the muscle pulling back the lower leg, but when outstretched, the quadricep is responsible for contracting, pulling the leg straight.
So, antagonstic muscle pairs aren’t antagonistic by definition, but still counteract the actions of the other muscle involved in the pair. Without antagonistic muscles, most animals would be able to move effectually.