The relationship between "green" and "blue" is an uncommon one: they are both types of colors. Since "sad" is an emotion, we must look for another word that is an emotion. The only choice that shares this relationship with "sad" is "angry." "Blue" is a color and unrelated in this regard. "Smile" and "frown" are facial expressions that accompany certain emotions. "Tears" can also be considered an expression of sadness, but is not an appropriate choice because "tears" are not an emotion themselves.

"Gold" and "silver" are both elements and precious metals, so to solve this analogy, you have to figure out which answer choice is grouped with "oxygen" in a similar fashion. Oxygen is an elemental gas, so the answer choice should also be a gas. The correct answer is therefore “nitrogen.” The answer cannot be "gas," "atmosphere," or "breathe," because although these words are related to oxygen, their relationship is not comparable to the relationship between "gold" and "silver."

"Fog" and "hail" are both varieties of weather, so we're looking for a word which fits into the same category as "red." While both "blue" and "color" may look like potentially correct answers, "color" is the category into which "blue" and "red" both fit, and "blue" an example of something that fits into that category along with "red," so "blue" is the correct answer, because as fog and hail are both types of weather, red and blue are both colors.