A sector is a portion of a circle—think of a “pie piece” or something of the like. Thus, the analogy could be rewritten, “Just as a sector is part of a circle, so is a segment part of X.” In general, a segment is any potential divided part of an object—in segmented worms, segments of cars, etc. Among the options provided, only “line” provides a suitable “whole,” to which the “part” that is a segment can be compared. All of the other options are broadly related but insufficient.

Sometimes an analogy can be so simple that you will overanalyze it. Do not do so with this question. Icing is merely meant here as the top layer on the cake—nothing further. Thus, the bridge sentence for this analogy could be written, “Just as icing is the top layer on a cake, so froth is the top layer of X.” Froth is the top layer of liquid caused by it being disturbed or sometimes by something like the fermentation of yeast in the liquid.

The ingredients of a recipe are the parts of which it is comprised; therefore, one could fairly write the bridge sentence, “Just as ingredients are the parts of a recipe, so are bristles part of a X.” Now, “bristle” can mean to become angry, but this comes from the description of an animal that is “bristling,” that is, having its hair stand up out of anger or fear. Bristles are a type of stiff hair that are used in brushes, which would be the best option for the “larger thing” of which the bristles are a part.

A district is a part of a given city. For instance, a city might have a “market district” in which a number of food salespeople have shops for selling bulk quantities to potential restaurants in town. There might be an “Italian district” in which a number of Italians live and have set up shops and restaurants, giving the area a distinctive Italian appearance and character. The analogy could thus be rewritten, “As a district is part of a city, so too is an hour part of X.” An hour is part of a day. While it is a period of time, a measure of time, a moment (in the grand scheme, at least), and even “a portion” (of a longer time), it is only part of a day. The analogy requires a part-whole relationship.

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.

An atom is the basic part from which molecules are constructed; therefore, this is a part-whole analogy. A minute is part of a larger measure of time. Here, you have to be careful. Two options are justifiably tempting, namely “hour” and “century.” Now, “molecules” are the “first rung” of combined atoms. We could, of course, say that organs are made up of atoms; however, this could be stated as “organs are made up of tissues, which are made up of cells, which are made up of molecules, which are made up of atoms.” Notice that “molecule” is the immediate product of atoms, not a distant one (as in the case of organs in relation to elements). This means that “hour” is a better option than “century.”

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.).

The word “nucleus” comes from the Latin for “kernel” or “nut.” In general, it can mean anything that is central in a given object. As you likely know from biology class, the nucleus is the name for the center of a cell. (It is, at least, “central” in a general manner. It does not necessarily need to be in exact center, of course). Just as a nucleus is the center of the cell, so is the “core” the center part of fruit, often containing the seeds.

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.