The paranormal realm is an area which is often neglected by scientific research, largely because by definition it encompasses concepts we cannot yet understand, and perhaps never will. Within the parentheses of the paranormal, however, lies a vast array of beliefs ranging from the thought-stimulating to the faintly ridiculous.
In the former category we could place, for example, the idea of alien life on other planets. The belief that perhaps some other planet in our gigantic universe contains species at some stage of evolution is one that is widely held; the possibility does not seem to be beyond the reaches of reason. The belief in angels, however, is wildly different from such a view.
That something is invisible does not mean it is not there: this seems to be a valid argument. It has been postulated, however, that the belief in angels is comparable to the belief in gravity, which is utterly ridiculous. Gravity is not a physical entity: it is a force. If Brad moves across the room at a certain speed, we say that we can see that Brad is moving. We cannot 'see' Brad's speed - not because it might not exist, but because it is not a 'thing' to be 'seen'. The same can be said for gravity; it is a the natural force of attraction between any two massive entities (that is, entities having mass).
Angels are not a force. Belief in angels is very different from, say, the belief in life after death; in gravity; in the laws of quantum physics; in evolution. To believe in angels - at least in the way that seems to be being put forward by the 'Real' side of this debate - means believing in literal, physical people in heaven. For a person who holds such a belief, attempting to persuade someone who has not formed their beliefs solely through having them dictated by a religious institution is difficult. Yes, the Bible is a valuable book. Yes, it contains stories of angels. Yes, if we take it as fact, angels exist. But many people do not take it as fact, and for very good reasons, upon which it is not necessary to elaborate here. Coming from an outsider's perspective, it is far easier to be persuaded to believe in almost anything than it is to be persuaded to believe in angels. It is possible, for instance, to construct an argument for the existence of fairies by stating that originally the 'Wee Folk' referred to a pygmy race separated from the rest of society, who became gradually mythologised until we arrived at the commonly accepted definition we have today. Personally, I can think of no such argument for angels; if anyone wishes to enlighten me, please do send a message.
Overall, the belief in angels seems to be difficult to justify without falling back on religious doctrine. If you happen to be a believer in angels, then they have a subjective existence in your reality. If, however, you are a non-religious person, it would be difficult, if not utterly impossible, to hold such a belief. Thus we arrive once again at the rift between the religious and the non-religious; which in itself is not as clear-cut as that sounds. The choice, if you believe in free will, is yours.