Yes.
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Yes. The sum of opposite angles is 180 degrees and that is a necessary and sufficient condition for a quadrilateral to be cyclic.
A square or a rectangle
Yes, every subgroup of a cyclic group is cyclic because every subgroup is a group.
every abelian group is not cyclic. e.g, set of (Q,+) it is an abelian group but not cyclic.
Only if it is a square.