They can be.
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Yes, two right angles can be adjacent to one another. In the letter "T", it's formed by two adjacent right angles.
A square because a parallelogram has two parallel sides and have two adjacent right angles but a square and a rectangle have also a two adjacent right angles.
Yes, they can be adjacent as well as supplementary.
The answer will depend on whether the two right angles are adjacent, alternate or opposite and also on the measures of the remaining angles.
Perpendicular.