The back muscles are antagonists during the pushup.
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When it comes to doing pushups, your chest, shoulders and triceps muscles do the work of making the motion happen. But they're not working in isolation: Your core muscles and even your thighs also contract to stabilize your body throughout the pushup motion. The role of antagonist muscles is less obvious. With the exception of antagonist stabilizers in your core, the "primary" antagonist muscles for a pushup are those that passively lengthen - effectively getting out of the way so that the primary movers (your chest, shoulders and triceps) can complete the pushup.
Identify the Players
When you do a pushup, three muscles kick in to make the movement happen. Your pectoralis, or chest, muscles generate the most force, swinging your arms in toward your midline. Your anterior deltoids - that is, the front fibers of your shoulder muscles - assist with this motion, while at the same time, your triceps muscles (at the back of your arms) contract to straighten your elbows. When you lower yourself back to the start of the pushup, all three muscles contract eccentrically, which means they lengthen under load, or, to put it another way, they act as brakes to control the downward part of the motion.
Know the Opposing Team
Because the three muscles already described - your chest muscles, your anterior deltoids and your triceps - work to generate the movement of a pushup, they're known as the agonist muscles for that exercise. The antagonist muscles for any exercise are those that perform the opposite motion from the agonists. So if your agonist muscles push, the antagonist muscles for that exercise pull, and vice versa.
For a pushup, the antagonist muscles are your back muscles (their "pull" motion is the opposite of your chest muscles' "push"); your biceps (they oppose the motion of your triceps); and your rear deltoids, or the back-facing fibers of your shoulder muscles, which oppose the motion of the anterior deltoids.
Understand the Role
Remember, your agonist and antagonist muscles perform opposing motions on any given joint. So if both agonist and antagonist muscles were to contract at once, the joint wouldn't move at all - it would be trapped in a tug-of-war of opposing forces, as it were. If you were to hold yourself in the "up" or "down" portion of a pushup without moving, both the agonist and antagonist muscles would be contracting at once.
If you're actually moving through the pushup, however, your antagonist muscles passively lengthen. This is how they "get out of the way" to let the agonist muscles do their job. The key exception to this is if you suddenly stop or brake your motion in the pushup. In that case, the antagonist muscles kick in to help your chest, shoulders and triceps control that sudden stop.
Keep it Stable
There is one more case in which your antagonist muscles kick in during pushups. Your abdominals and quadriceps contract to help keep your body in the plank position throughout the pushup - but if they contracted on their own, you'd actually end up in a pike position. So your glutes and erector spinae - a strong stabilizer along your back - also contract. As the antagonists to your abs and quadriceps, your glutes and erector spinae help hold your body stable throughout the exercise.