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which answer identifies the participial phrase and the word it modifies in this sentence?
the dog barking in the yard for the last three hours is giving me a headache.

the participial phrase "for the last three hours" modifies "barking". <
the participial phrase "in the yard" modifies "barking".
the participial phrase "barking in the yard" modifies "dog".
the participial phrase "giving me a headache" modifies "hours".

The 3rd option, The participial phrase "barking in the yard" modifies "dog".

Explanation:

A participal phrase consists in a participle (past participle or present participle <verb+ing>) and a complement to the participle (nouns, pronouns, direct or indirect object, etc.). These phrases fuction as adjectives, it means, are intended to modify a noun or pronoun in the sentence.

In the sentence, "The dog barking in the yard for the last three hours is giving me a headache."

Barking is the participle (verb bark + ing)

in the yard for the last three hours is the complement of the participal phrase

The phrase is explaining why the dog is giving me a headache (because the dog has been barking in the yard for the last three hours), so is directly connected to the noun "The dog"  modifying it.

the president was the speaker and it was a great part of history.

explanation:

video of the national aeronautic space administration's (nasa) coverage of president john f. kennedy’s address at rice university, houston, texas, concerning the nation's efforts in space exploration. in his speech the president discusses the necessity for the united states to become an international leader in space exploration and famously states, "we choose to go to the moon in this decade and do the other things, not because they are easy, but because they are hard."

the answer is fantasy