This analogy is relatively simple. A pocket is part of a shirt, so you are looking for the that of which a lace is part. While lace can be a type of plaited fabric, here, the best option is to understand it as meaning the long string used in things like shoes. The lace is part of a shoe as a pocket is part of a shirt. 

A "link" is one of the small metal pieces in a longer chain. It also can mean any portion of a long “chain” of things (understood metaphorically). Hence, we speak of the “missing link” between apes and men—that is, the missing intermediary stage / being in the great chain of beings. The analogy could be written, “As a link is part of a chain, so is a chapter a part of X.” “Book” is the only option of something of which a chapter is a portion. (A chapter, of course, could be a part of a thesis, a novel, a report, etc.).

A citizen is a member of a nation and could justly be called a part thereof. Our bridge sentence could thus be written, “As a citizen is a part of a nation, so is a branch a part of a X.” Among the options provided, only “tree” gives the larger thing of which a branch is a part.

The hinge is the part of the door that lets it swing. Similarly, the fulcrum is the part of the lever that lets the lever arm swing.

“Dregs” are defined as either “the remnants of a liquid left in a container; sediment, grounds” or “the most worthless parts of something.” So, the bits left in the bottom of a glass after one drinks tea are called “dregs,” and we need to pick out an answer choice that is a noun conveying what might be left in the bottom of a glass after drinking coffee. While “grounds” and “beans” are both words associated with coffee, coffee is made from beans, but one does not find “beans” in the bottom of one’s empty coffee mug, so “beans” cannot be the correct answer. “Grounds,” however, is a noun that can mean “solid particles, especially of ground coffee, that form a residue; sediment.” So, because tea might leave dregs in the bottom of a glass while coffee might leave grounds in the bottom of a glass, “grounds” is the correct answer.

The outer "framework" of an animal's body is its skin, which protects its insides. A similar relationship exists between an orange and its peel, since the peel is the outer part of the fruit which protects the inside.

A knob is a part of a door; therefore, you could translate this analogy as, “Just as a knob is part of a door, so is a burner a part of a X.” A burner is the heating surface on a stove. This is the only option that functions as a whole of which the burner is a part.

A "jogger" is one part of a marathon, indicating a part to whole relationship. Both words are also nouns. "Religion" is also a noun, meaning the missing word is a noun and part of religion as a whole.

"Creed" (noun) is the correct answer, root word "cred" = believe. It is a set of beliefs that are part of a religion.

"Polytheistic" (adj) prefix "poly" = many, root "theo" = religion, gods. This is a type of religion, not a part of religion itself.

"Cult" (noun), a group of people sharing (usually fringe) beliefs. This is related to the definition of religion, but not part of one.

"Pious" (adj) someone who is highly religious. People are parts of religion, but this word describes a particular individual, not an object or part.

"Immoral" (adj) prefix "im" = not, root "mor" life. This word also describes someone or something as being not right or ethical, and it is not an object or part.

A "compass" is used to find direction in the act of navigation (the plotting or directing of a course for a ship or vessel). While this may seem like a tool-user relationship, it is not. The compass is part of the whole subject of navigation. Likewise, the answer will be part of a gemstone.  As all the words in the analogy set have been nouns, the answer should also be a noun.

The correct choice is "facet" (noun), which is one side of a multi-sided thing. Think of it as one of the flat sides of a jewel.  Its root "face" literally means face, appearance, or side. This is similar to how sides of 3D geometric shapes are often called "faces."

"Bauble" (noun) is a trinket or piece of jewelry, which makes it more a synonym of a gemstone.

"Peridot" (noun) is a type of gemstone, olive-green colored.

"Scarce" (adj) means uncommon, special, or rare. This describes the gemstone, rather than relating a part of one.

"Entombed" (adj) means buried, generally with a body, like in a grave or tomb. It was a common practice to bury wealth and treasures with someone of importance, but again, the word does not relate an example of a part of a gem.

A "battalion" is a group of soldiers within a larger army, something like a "regiment" or a "corps." So, to solve this analogy you need to determine which of these is a group of individuals who are collectively a part of a "company." The correct answer is a "branch" which means a regional department of a large company.