No they are two entirely different things.
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No, it can never be. One is an angle and the other is a curved line!
Only if the congruent angle is the angle between the two congruent sides (SAS postulate).
No, because they need not be congruent.
If the parallelogram is a square then angle A is congruent to angle B ,is congruent to angle C. AB is congruent to BC is congruent to CD.
Well, in degrees, the arc is congruent to its central angle. If the radius is given, however, just find the circumference of the circle (C=πd). Then, take the measure of the central angle, and divide that by 360 degrees. Multiply the circumference by the dividend, and you will get the arc length. This works because it is a proportion. Circumference:Arc length::Total degrees in triangle:Arc's central angle. Hope that helped. :D