Teeth is to Hen as Nest is to which of the following, based on common English idioms?

Introduction / Context:
This analogy is not about simple physical characteristics but about well known idiomatic expressions in English. The pair Teeth and Hen hints at the saying hen teeth, which is used to talk about something extremely rare. You must link Nest with the correct word from the options to form a similarly famous idiom.

Given Data / Assumptions:
1) Hens do not literally have teeth.
2) The phrase hen teeth is used idiomatically for something that is almost impossible to find.
3) Nest combined with a certain animal in everyday usage refers to a misleading or imaginary discovery.
4) The options are Gums, Lips, Mare, and Mouth.
5) The question expects knowledge of idioms, not just physical facts.

Concept / Approach:
The first pair Teeth and Hen reminds us of the idiom as rare as hen teeth, meaning extremely rare or non existent. It combines a body part that birds do not have with the bird to create an impossible image. The second part of the analogy should echo this type of expression. There is a well known English idiom mares nest, which refers to a supposed discovery that in reality is worthless or imaginary. It similarly uses a combination that is not literally correct, since mares do not build nests. Therefore, NEST must be paired with MARE to form this idiomatic expression.

Step-by-Step Solution:
Step 1: Recognise that hens and teeth do not go together naturally, which suggests an idiomatic or humorous phrasing rather than a simple physical relation. Step 2: Recall the idiom about hen teeth, which is used to describe something almost impossible or extremely rare. Step 3: Look for a similar idiom in English that uses Nest with a particular animal. The expression mares nest describes a supposed discovery that actually turns out to be illusory or not valuable. Step 4: Notice the pattern. Hen teeth suggests impossibility. Mare nest suggests a false or foolish discovery. Both use combinations that are not literally accurate in the real world. Step 5: Check each option. Gums and Lips are body parts, and Mouth is another body part. They do not form any common idiom with Nest that parallels hen teeth. Step 6: Mare, on the other hand, forms mares nest, which is a known idiom. Thus, Nest is to Mare as Teeth is to Hen in the context of idiomatic usage.

Verification / Alternative check:
Search your memory of idioms. As rare as hen teeth and mares nest appear in many idiom lists and advanced vocabulary references. There is no standard idiom gums nest, lips nest, or mouth nest in English. This confirms that the intended testing point is knowledge of these two specific idioms.

Why Other Options Are Wrong:
Gums, Lips, and Mouth all refer to parts of the mouth but they do not combine with Nest to produce a meaningful idiom. In addition, they do not mirror the pattern of using an animal name to create a phrase about impossibility or folly. Only Mare meets both requirements: it forms a known idiom with Nest and mirrors the animal body part pairing style of the first pair.

Common Pitfalls:
Students who are not familiar with idiomatic usage may try to interpret the question literally and think about where nests are found, leading them to random guesses. However, hens do not have teeth, so the first pair already signals an idiomatic angle. Recognising this quickly points your thinking toward matching idioms rather than purely physical features.

Final Answer:
The correct completion of the analogy is Nest is to Mare in the same idiomatic sense that Teeth is to Hen. The answer is Mare.