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A trapezoid.
Yes, that sounds right.
Angles that are congruent and supplementary must be right angles.
Only in squares and rectangles. In a rhombus, the consecutive angle is supplementary (sums to 180 degrees).
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.