Hi, I think your question is asking how virtue is being used in each of the sentences.
For the first one, "He virtuously helped the old lady across the street", virtue is being used as an adverb. An adverb is a word that describes the manner in which a verb is being done. Examples of adverbs are "slowly", "clumsily", "quickly", etc. A trick to finding adverbs is looking for suffixes like "-ly" at the end.
For the second sentence, "The virtuous knight proved worthy in battle", virtue is being used as an adjective. An adjective is a descriptive word that describes a noun, which is a person, place, or thing. Examples of adjectives are "fast", "smart", and "cold".
For the third one, "Her virtuousness was widely respected", virtue is used as a noun. A noun is the subject of the sentence and a person, place, or thing. Virtuousness is the subject of this sentence and is the thing that the verb is affecting, so I think it is being used as a noun because it wouldn't be any of the others.
I hope this helps you! :)
