Not always. If two angles are congruent then they simply have equal measure.
They must only be right angles if they are supplementary, that is, they must both add up to 180 degrees.
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The theorem states "If two angles are both supplementary and congruent, then they are right angles."
Only the two angles which are connected by the shorter diagonal will be congruent. The other two angles will not necessarily be congruent.
Two angles are said to be congruent if they are equal.
A parallelogram cannot have only two congruent sides, nor only two congruent angles.
Of course not. The right angles at the corners of my book are certainly not vertical with the right angles at the corners of your computer screen, but they're congruent. The "if" is true, but the "only if" is not. Verticality is sufficient but not necessary for congruence.