"Stubborn" and "hardheaded" are both synonyms, just as "laid-back" and "easygoing" mean the same thing.

"Alabaster" is often used to describe "white," just as "jet" is often used to describe "black."

A "river" is filled with "fresh" water whereas an ocean is filled with "saline" (or salty) water.

Something “frivolous” is not serious, and when applied to a person, the term can mean carefree. Frivolous pleasures are really shallow, inconsequential pleasures. Thus, we could say, “Just as frivolity is shallow, so is gravity X.” Although “gravity” is often used in the sense of a physical force, it can likewise mean “weighty” in a metaphorical sense, as in “a grave, important manner.” Someone who has gravity (or as is often said in the media, “gravitas”) has a certain solemnity—or, “is solemn.”

An atrium is a type of entranceway that has an open ceiling, so the analogy must be referring to the ceiling when it places “open” in relation to “atrium.” The best way to write a bridge sentence would be, “As the ceiling of an atrium is open, so is the ceiling of a greenhouse X.” None of the other options pertain to the ceiling of a greenhouse. Even if the option “closed” were also among the other answers, even then would “glass” be the best answer, for it is more specific to “greenhouse” than “closed.” The atrium’s open roof / ceiling is a necessary characteristic. This analogy is not opposing closed and open as much as it is paralleling the characteristics of the two types of structure.

The word “credence” means belief in the truth of something. If someone is "trustworthy," we are likely to give credence to his or her words or claims. Something "malicious" is evil or ill-intentioned and harmful. Such a thing or person deserves to be feared, which adequately fulfills the parallel for this analogy.

A nebula is a cloud of dust and gas in space, appearing misty like dust in the air—at least when magnified by a telescope. The word actually comes from the Latin word for “mist,” and when we call something “nebulous,” we imply that it is unclear and / or vague, as though it were shrouded in mists. The analogy could be written, “As a nebula is misty in appearance (or misty in character), so is a vacuum X in appearance / character.” Note that the analogy requires some flexibility since the two words are not quite the same in character—a vacuum really is a lack of something, one could say somewhat paradoxically, “It is a nothing.” In any case a property of being a vacuum is the fact that it has no contents and thus is empty.

The cardiologist is a specialist at studying the heart. The word itself as a root for “heart” in it—namely, “card-.” A “cardiac arrest” occurs when the heart suddenly stops beating. An “ecologist” is one who studies the environment, generally speaking. An ecosystem, for example, is the whole system of organisms and material environment in a given area. While each of the other options are part of the environment and animals therein, none of them is equivalent to the environment in general.

A musician performs a concert, so the answer must be something that a dancer performs. While you might think that “gyration” would work, this is a specific action, not a type of event (like concert). Thus, “ballet” is the best answer for this. Even though all dancers do not necessarily perform ballets, this option alone establishes a parallel with dancer in a manner similar to the relationship between musician and concert.

"Cardiovascular" is a medical term relating to the "heart," just as "pulmonary" refers to the "lungs."