No, they need not be.
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No. If two angles are congruent they have the same measure. But that measure can be anything.
Yes, in similar triangles, the angles are always congruent, and the sides have the same proportions to each other.
Are Congruent.
No, a pair of angles that are supplementary will always have a sum of 180 degrees, while a pair of angles that are congruent will have the same measure. Therefore, it is not possible for a pair of angles to be both supplementary and congruent.
Give us a break! -- A 3° angle is congruent to another 3° angle, but their sum is only 6° , not 180°. -- Congruent angles are always equal, but supplementary angles don't have to be equal.