This sentence contains a dangling participle. A dangling participle is a participle in an introductory phrase that makes reference to the wrong noun. As written, it appears as if "the sun" is doing the "noticing," when "the children" should be the ones doing the "noticing." Rearranging the sentence so that "the children" is the noun that follows the introductory phrase rids the sentence of its dangling participle.

The phrase "Rounding the corner" is a type of error is known as a "dangling modifier." What is described as "rounding" is not "the mountains" as the sentence's construction implies by having "the mountains" immediately follow the introductory phrase "Rounding the corner." "The group" is the noun being described by "Rounding the corner," so to correct this sentence's ambiguity, the sentence should be arranged so that "the group" is the noun that immediately follows "Rounding the corner." Thus, the correct answer is "The mountains loomed large in front of the group as they rounded the corner."

When the sentence begins with a descriptive phrase (e.g. prepositional or adverbial), the subject of that phrase should immediately follow. Only two choices correctly place the Subject, but one omits the primary verb, creating a new structure issue.

This sentence contains a dangling participle. The way the sentence is phrased, the introductory phrase "Gazing into the night sky" appears to describe "the stars" instead of "Galileo." The only answer choice that corrects this issue while creating a grammatically correct sentence is "Gazing into the night sky, Galileo was prompted by the stars."

The sentence is written in a manner that makes it appear that "the machine" was "working out the kinks," when the only logical noun to do this action is "the tinkerer." The sentence needs to be rearranged so as to indicate clearly who was doing the "working," a task best achieved by the answer choice "the tinkerer exhibited the machine in 1904."

The sentence as written features a dangling modifier, which makes the sentence read as though "the breeze" was "walking out of the house." The sentence needs to be rearranged to make it clear that the girl was the one doing the walking. The only answer choice that does this is "The breeze nearly knocked the girl down as she walked out of the house."

The sentence is written in a way that makes it appear that "the smog" was the thing "flying into the city." The sentence needs to be restructured to show that the plane, and the people on it, were "flying into the city." The answer choice that best clarifies the meaning of the sentence is "The smog was visible to everyone on the plane flying into the city."

The sentence is incorrect because it has a dangling modifier. The thing “working diligently in the garden” is the farmer, not the weeds, so it should start with “the farmer” directly after the comma. Also, “removed” is the most precise and least redundant verb tense to use. 

The sentence as written contains what is known as a "dangling modifer," as the introductory clause does not describe action taken by the subject, "noises." This means the sentence needs to be fixed to make the subject fit with its modifying clause. The best choice among the answers is "Walking at night through the graveyard, people hear noises that seem."

The modifier is "after eating a pan of cake." The only conceivable thing that could eat a pan of cake is Jimmy, not his stomach. So, the initial modifying phrase needs to be placed next to "Jimmy" in the sentence so that it is clear to the reader that this is what is happening. This means that "Jimmy" needs to immediately follow the sentence's initial modifying phrase. "Jimmy, after having ate a pan of cake, felt ill" is incorrect due to the incorrect verb construction "having ate." The correct verb phrase would be "having eaten."