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A trapezoid.
There are instances that quadrilateral angles can be consecutive and opposite angles are congruent. The best examples are square and rectangle.
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.