A bird has a certain mass. A cage also has a certian mass. The air within the cage, too, has a certain mass. In a closed system(air-tight cage)the mass, and thus the weight, of the bird will remain the same whether in flight or perched. The downward force of the bird has to be the same as the weight of the bird to keep it aloft. True, some force may be lost at diagonal and horizontal vectors, but ultimately if the bird is to remain in the air there must me a downward force equal to it's weight that must be applied to the system. It's the same as when he was on the perch. The bird was applying his weight(force)on the perch and the perch exherted an equal force on the bird. That is what kept the bird from falling through the perch. Likewise, when the bird is in flight it is beating it's wings in an effort to transfer it's weight onto the air. The air pressure created by the flapping of the bird's wings exherts an equal pressure on the underside of the bird's wings so that the bird will remain suspended in the air. All this is to say that, no matter what suspends the bird in the cage, be it the air or a perch or even the base of the cage, the same forces are applied to the system as a whole. As long as mass is not lost in the transfer of weight from perch to air then total weight of the system will remain the same.

Shan-My brain hurts-dar