Step 1: Understanding the Question.
The given sentence is a conditional statement of the form "if it is raining, then peacocks dance". We need to find the option that must be true whenever this original statement is true, using nothing beyond the sentence itself.
Step 2: Key Formula or Approach.
Write the statement as "raining implies dancing". For any conditional "P implies Q", only one other statement is guaranteed to carry the same truth value: the contrapositive, "not Q implies not P". The converse ("Q implies P") and the inverse ("not P implies not Q") are not guaranteed to be true just because the original statement is true.
Step 3: Detailed Explanation.
Let P be "it is raining" and Q be "peacocks dance". The given sentence says P implies Q.
The contrapositive of "P implies Q" is "not Q implies not P", which in words is "if peacocks are not dancing, then it is not raining". This must always hold whenever the original sentence holds.
(A) Peacocks dance only when it is raining: this says dancing implies raining, which is the converse of the original statement. The sentence never claims peacocks dance for no other reason, so this is not guaranteed.
(B) When peacocks dance, it is raining: this is the same converse statement as (A) in different words, so it is also not guaranteed.
(C) When peacocks are not dancing, it is not raining: this is exactly "not Q implies not P", the contrapositive of the original sentence, so it must be true.
(D) When it is not raining, peacocks do not dance: this says not P implies not Q, which is the inverse of the original statement, not the contrapositive, so it is not guaranteed either.
Step 4: Final Answer.
The statement that is necessarily true is option (C).
\[ \boxed{\text{When peacocks are not dancing, it is not raining.}} \]
