Which of the following is an appropriate way to check for responsiveness?

An appropriate way to check for responsiveness is to tap the person lightly on the shoulder. This method is effective because it allows the responder to assess the person's state in a gentle manner, which is crucial when determining if someone is conscious and aware without causing further distress or harm. Tapping gently does not startle the individual unduly and is a common practice in first aid scenarios.

The other options may not produce the desired outcome. Yelling loudly at someone could inadvertently create panic or fear, which may not be conducive to an accurate assessment of responsiveness. Blowing air in their face is not a standard or recommended practice and may be considered rude or invasive. Shaking their arm violently is also inappropriate, as it could cause discomfort or injury, particularly if the person is unconscious or in a vulnerable state. Proper assessment should always be done with care and consideration for the individual’s wellbeing.