1. I am sure that she was angry with him.
2. I am sure she was angry with him.
3. I am certain that she was angry with him.
4. It is sure/certain that he was pleased with the gift.
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Can we use both 'sure' and 'certain' in the sentences? In Sentence 4, the 'that clause' is a noun clause. Then, what about the 'that clause' in Sentences 1 and 3? A noun clause or an adverbial clause?
Yes, "sure" and "certain" mean the same thing here.
In these sentences, "that" is a subordinating conjunction, I think. Another tutor may correct me!
http://grammar.ccc.commnet.edu/grammar/conjunctions.htm#subordinating_conjunctions
and
http://grammar.ccc.commnet.edu/grammar/conjunctions.htm#that
In sentences 1, 2, and 3, both "sure" and "certain" can be used interchangeably to indicate a strong belief or confidence in the statement that follows. Therefore, all three sentences convey the same meaning.
Regarding the clause introduced by "that" in sentences 1 and 3, it is a noun clause. A noun clause functions as a noun, acting as the subject or object of a sentence. In both sentences, the clause "that she was angry with him" serves as the direct object of the verb "am sure" (in sentence 1) or "am certain" (in sentence 3).
In sentence 4, although "sure" and "certain" can both be used, the "that clause" functions as an adjective clause modifying the pronoun "it." Adjective clauses provide additional information about a noun or pronoun.
To determine the function of a clause, you can ask yourself, "What role does this clause play in the sentence?" If the clause acts as a subject or object, it is likely a noun clause. If it describes or provides more information about a noun or pronoun, it is most likely an adjective clause.