"Relieved that the election was finally over, the governor's celebratory dinner was incredibly lavish." - This option is incorrect due to a dangling modifier. The first clause, "relieved that the election was finally over," refers to "the governor," therefore "the governor" must be the subject of the second clause, rather than the celebratory dinner (which, since it is not, to our knowledge, sentient cannot feel relief, nor even the stress that would precede such relief). 

"Relieved and celebratory that the election was over, the governor's dinner was incredibly lavish. " - This option includes creates the same dangling modifier as did the previous option, alters the meaning of the sentence, and incorrectly uses "celebratory" as an adjective describing a mood (it would be correct to say "feeling celebratory").

"Relieved that the election was over; the governor's celebratory dinner was incredibly lavish." - This option keeps the same dangling modifier error, while adding a semicolon error. The first clause of this sentence is a dependent clause and therefore cannot be separated from the main clause with a semicolon.

"Relieved that the election was over, he the governor's celebratory dinner was lavish." - While still repeating the dangling modifier error, this opiton also unnecessarily, and confusingly adds the pronoun "he" to the main clause, even thought "the governor" is a clear and obvious label for the subject.

The correct version of this sentence reads: "Relieved that the election was finally over, the governor held an incredibly lavish celebratory dinner." - Making the governor the subject of the main clause is the best way to fix this dangling modifier.

The example sentence includes a dangling modifier. The introductory phrase "Traveling by plane," must be immediately followed by the subject that was, in fact, traveling by plane. Only two of the provided options correct this dangling modifier error. Of those two it is more concise, and clear, to say "we saw roads and highways that looked like thin, winding chords," rather than "we saw, like thin winding cords, the roads and highways."

In this sentence, the section that modifies the subject—“unlike the modern model of the atom”—must proceed the main clause, rather than succeed it. "Unlike the modern model of the atom, the Bohr model states that electrons travel in fixed paths around the nucleus," fixes the problem.

In the sentence above, as it is written, it is not clear what the modifying phrase "After he returned from the long walk" is referring to. A chair cannot walk. In order to avoid a dangling modifier, a subject which can be appropriately modified by such a phrase must be included. The best way to change the underlined portion of the sentence above is: "the man settled into the chair on the porch."

The sentence above contains an example of a squinting modifier. It is not clear if the word "always" is meant to modify how often the young people listen to their elders, or how often they end up in trouble. This error must be corrected by changing the position of the modifier. The best way to correct the underlined portion of the sentence above is: "Young people who don't always listen to their elders wind up in trouble."

In the example sentence, it is not clear what the modifying phrase "Thrilled that the storm had finally broken" is referring to. Boots cannot be thrilled. In order to avoid a dangling modifier, a subject that can be appropriately modified by such a phrase must be included. The best way to change the underlined portion of the sentence above is: "the campers marched gleefully along the trail."

The sentence above contains an example of a squinting modifier. It is not clear if the word "vigilantly" is meant to modify how the old folks keep track of their health, or how they die from simple health complications. This error must be corrected by changing the position of the modifier. The best way to correct the underlined portion of the sentence above is: "Elderly people who don't vigilantly keep track of their health die from simple health complications."