Final answer:
The speaker used the Pathos appeal to persuade her mother to get a puppy by playing on her emotions, making her feel both sympathetic and excited about the prospect of a new pet.
Step-by-step explanation:
The scenario provided describes a speaker using a rhetorical appeal to influence her audience, which in this case, is her mother. From the given options, the appeal used is Pathos; by appealing to her mother's emotions, the speaker made her mother feel compassionate about not having a puppy and excited about the joy of having a cuddly friend. This type of appeal is known to evoke emotions such as sympathy, happiness, or excitement, thereby persuading the audience through a shared emotional experience.