Two angles are said to be congruent if they are equal.
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A parallelogram cannot have only two congruent sides, nor only two congruent angles.
Only the two angles which are connected by the shorter diagonal will be congruent. The other two angles will not necessarily be congruent.
No. Angles are not congruent. (Triangles may be congruent.)
Yes.
No. A square has two pairs of congruent angles!