Answer:
1. The idea of alien life (proposed many years ago) has not yet been proven- Participle phrase.
The participle phrase starts either with the present (dependably ending with -ing) or past (dependably ending with -ed)
The participle phrase includes the modifiers or objects to complete the thought of the sentence.
2. (According to the letter) the wedding is June 25th.- Prepositional.
A prepositional phrase refers to the phrase which begins with a preposition and ends with the 'objects' of the prepositions that are nouns, gerund, clause or pronoun.
The pattern of the prepositional phrase follows by the object of the preposition having one or more modifiers.
3. Mikayla's homeroom teacher, (mr. Jones) is very proud of her- Appositive
Appositive refers to a word or a noun that is followed by another phrase or noun which identifies or rename the noun or word. In other words, it refers to the literary device which appears after or before a noun phrase or noun, used with a comma.
4. (Talking) is not allowed in class- Gerund
Gerund in a sentence functions as a noun. Hence in a sentence, a gerund will act as direct objects, subjects, subject complements, an object of prepositions or indirect objects. Every gerund ends in -ing, and so does present participles. On the other hand, present participles act as modifiers or complete progressive verbs.
Explanation: