The correct answer is option 2: "They share rent of the house equally."

This sentence requires the definite article "the" because it refers to a specific house. The other sentences refer to general concepts or locations. Let's examine why the other options are incorrect:

• Option 1: "He lives somewhere in southern Germany." Here, "southern Germany" is a general region; it doesn't refer to a specific house or location within southern Germany. Thus, "the" is not needed.
• Option 3: "She is ignorant of her social standing." This sentence discusses a general concept—social standing—not a particular instance. Therefore, "the" is not required.
• Option 4: "She has an abundance of cake at home." Similar to the other incorrect options, this sentence describes a general quantity of cake, not a specific, defined quantity. The article "the" is not grammatically necessary.
• Option 2: "They share rent of the house equally." This option requires "the" because "house" refers to a specific dwelling, not just any house. The definite article "the" is used when talking about a specific item or a previously mentioned item. In this case, the sentence implies a particular shared house, implying a specific shared space.

Understanding the use of definite and indefinite articles is crucial for proper English grammar. The choice between "a," "an," and "the" depends on whether the noun is specific (requiring "the") or general (requiring "a" or "an").