The phrase "are indignant" conveys feelings of anger, frustration, and outrage. Synonyms for this phrase could include "are incensed", "are outraged", "are irate", "are furious", or "are annoyed". These words all convey a strong emotional response to a particular situation, and indicate a level of anger or frustration that can be difficult to contain. To be indignant is to feel a sense of righteous anger, and these synonyms all convey a similar sentiment. When someone is indignant, they are not just angry for the sake of being angry - they are angry because they feel that a wrong has been committed, and that justice must be served.