Sometimes. For example, a rectangle has one set of four equal angles, and a parallelogram has two sets of two equal angles.
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Yes, the opposite angles in a regular quadrilateral are equal.
There is no specific name for such a quadrilateral.
Any 4 sided shape is a quadrilateral including a a square which is a regular quadrilateral because it has 4 equal sides and 4 equal angles
Diagonally opposite pairs of angles will always be equal, but the only time all angles in a quadrilateral parallelogram are equal is when that shape is a rectangle.
It is a rhombus that has 4 equal sides but no right angles.