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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.
That depends on what type of a 4 sided quadrilateral it is but in generals the 4 interior angles of any quadrilateral always add up to 360 degrees.
No. For any random quadrilateral, they are actually "never" congruent. The probability of the sides being so is infinitely less than the probability they will not be.
There are instances that quadrilateral angles can be consecutive and opposite angles are congruent. The best examples are square and rectangle.