1. Write the names of the four parts of an English letter using the words in the box.

(What does 'using the words in the box' modify? Is it an adjective phrase?)

2. Look at the photos and complete the sentences using the relative adverb, where.
(What does 'using the relative adverb, where' modify? Is it an adjective phrase?)

1. "using the words in the box" is telling HOW to write the letter. Therefore, it's adverbial, modifying the main verb "Write."

2. "using the relative adverb" is also telling HOW -- this time it's modifying the main verb in the second clause, "complete."

Adverbs can be single words, or they can be phrases or clauses. Adverbs answer one of these four questions: How? When? Where? and Why?
http://www.chompchomp.com/terms/adverb.htm

Thank you for your help.

Then don't we need to put a comma as follows?

1. Write the names of the four parts of an English letter, using the words in the box.

2. Look at the photos and complete the sentences, using the relative adverb, where.

1. "Using the words in the box" is a prepositional phrase modifying the verb "write." It clarifies that the names of the four parts of an English letter should be written utilizing the words provided in the box.

2. "Using the relative adverb, where" is also a prepositional phrase modifying the verb "complete." It specifies that the sentences should be completed by incorporating the relative adverb "where" in order to describe the location shown in the photos.