Select the most appropriate ANTONYM of the underlined word.
The brave knight set out to vanquish the fearsome dragon and protect the kingdom from its menace.

The correct option is Liberate.

Let us break down the meaning of the words to understand why this is the correct antonym:

1. Analyze the underlined word:
The word vanquish means to defeat thoroughly, conquer, or bring under complete control (e.g., to defeat an enemy or opponent in a battle or contest). In the context of the sentence, the brave knight wants to conquer or defeat the dragon.

2. Define the antonym (opposite meaning):
We need to find a word that represents the opposite of conquering, defeating, or subduing. The opposite of subjugating or conquering someone/something is to set them free.

3. Evaluate the options:
- Liberate: This means to set someone or something free from imprisonment, slavery, or enemy occupation. Since "vanquish" involves conquering or subduing, "liberate" (setting free) is its direct opposite (antonym).
- Measure: This means to ascertain the size, amount, or degree of something. It is unrelated to defeating or freeing.
- Engage: This means to participate or become involved in something, or to attract/hold someone's interest. It is not the opposite of vanquishing.
- Instill: This means to gradually but firmly establish an idea or attitude in a person's mind. It is not related to defeating or freeing.

Therefore, the most appropriate antonym of vanquish is Liberate.