The participial phrase running after her dog is made with the help of the present participle of the verb run and functions as an adjective modifying ‘she’.
A participial phrase consists of a participle, modifiers and complements.
Running after her dog is a participial phrase consisting of the participle running and the prepositional phrase <em>after her dog, </em>which includes the preposition <em>after,</em> the object of the preposition <em>dog</em>, and the possessive adjective <em>her</em>.
The reason why the Declaration of Independence is so persuasive is that Jefferson uses deductive arguments, which includes examples of Britain's wrong doings, which gives the person reading the document a chance to see exactly why they are breaking from Great Britain.
The first thing I think of when somebody says american dream is fame and wealth but since you said that i can't write fame or money as the answer I think that everyone gets equal chances or opportunities to succeed in life and achieve goals.