The word 'apple' begins with a vowel sound, so 'an' is required. It is mentioned for the first time, making the indefinite article suitable. 'A apple' would be grammatically wrong. 'The apple' would mean one already known in the context. Hence, 'an apple' is correct.

Countries that are plural or contain descriptors take the definite article 'the'. The phrase 'United States' falls into this category. 'A United States' or 'an United States' would be ungrammatical. 'No article' would also be incorrect here. Therefore, 'the United States' is correct.

The adjective 'interesting' begins with a vowel sound, so 'an' is correct. This is the first mention of the book, making the indefinite article suitable. 'A interesting book' would be grammatically wrong. 'The' would suggest a particular one already known. Thus, 'an interesting book' is the right choice.

The definite article 'the' is used to point to a particular man visible or already known in the context. 'A man' would mean any man, not a specific one. 'An man' would be grammatically incorrect. 'No article' would leave the sentence incomplete. Hence, 'the man' is correct here.

The stars in the sky are a universal reference, so the definite article 'the' is needed. This signals that the listener knows which stars are being referred to. 'A stars' would be grammatically incorrect. 'No article' would also leave the meaning incomplete. Therefore, 'the stars' is correct.

The noun 'engineer' begins with a vowel sound, which requires 'an'. Indefinite articles are used when introducing something not mentioned before. Saying 'a engineer' would be grammatically wrong. If it were a particular engineer known in context, 'the' would be correct. Thus, 'an engineer' is right here.

The noun 'hotel' begins with a consonant sound, so 'a' is correct. Since the hotel is being mentioned for the first time, the indefinite article is suitable. 'An' would be wrong because the pronunciation starts with a consonant. 'The' would only apply if it referred to a particular hotel already known. Thus, 'a hotel' is correct here.

The adjective 'old' begins with a vowel sound, so 'an' is correct. This is the first mention of the house, which makes the indefinite article suitable. 'A old house' would be incorrect due to pronunciation rules. 'The old house' would suggest one already known in context. Therefore, 'an old house' is correct.

The adjective 'beautiful' begins with a consonant sound, so 'a' is required. The song is mentioned for the first time, which makes the indefinite article correct. 'An' would not fit here because of pronunciation. 'The' would suggest a specific song already known. Hence, 'a beautiful song' is the right choice.

Unique monuments and famous places always take the definite article 'the'. The Pyramids of Giza are a specific and world-famous landmark. 'A Pyramids' or 'an Pyramids' would be incorrect. The definite article emphasizes their uniqueness. Thus, 'the Pyramids of Giza' is correct.