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Your teacher is right , but simplified the explanation by not explaining fully what he means by radiation. You highlighted the holes in his explanation. Heat energy is transferred by either radiation or convention. This is what I was taught at age 15, a very long time ago. Convection can only occur via an atmosphere or other material whereas radiation requires no such medium. As you have discerned,
we then have to ask what drives radiation ( photons - gamma rays to microwaves ) and what drives convection ( molecular vibration - at absolute zero vibration ceases ).
You have an enquiring mind, which would be ideal for a career in science. Rather than giving you detention, I would have encouraged your enquiry, bringing in the rest of the class.
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The sun gives off electromagnetic radiation in the form of UV, infra red and of course visible light. Now electromagnetic waves go faster through a vacuum. We all know the UV rays damage our skin. Once the rays pass through the atmosphere some of the energy is then trapped, this causes heat.
You are both right on this, don't disagree with a teacher just say to yourself I was right and smile :)