For protection from the summer sun, the Mojave lived in open-sided, flat-topped dwellings known as shades, each a roof of poles and arrowweed supported by posts set in a rectangle.
(A) each a roof of poles and arrowweed
(B) each a roof of poles and arrowweed that are being
(C) with each being a roof of poles and arrowweed
(D) with roofs of poles and arrowweed to be
(E) with roofs of poles and arrowweed that are
B and C are out because of 'being'.
In D, ' to be ' is redundant.
E comes close but ' that ' incorrectly modifies ' arroweed '. It should modify ' shades '
always refer to the thing immediately preceding it.... or it can refer to a subject much before in the sentence.
That ALWAYS refers to the subject. That's why prepositional phrases (that have nouns disguised as subjects as in this example - poles and arroweed) immediately followed by a relative "that/which" create a huge confusion - however, once you mark the prepositional in parenthesis you will be able to see the real subject -
So in this example - "that" still modifies "shades"
shades (with roofs of poles and arroweed)
that are supported by posts set in a rectangle.
However there is still a problem with the choice E.
The sentence in E looks as if "known as shades that are supported by posts set in a rectangle" - however it is not the shades themselves that are supported by posts - rather, as only the choice A brings out - these shades are in turn structures that are "each a roof of poles and arroweed supported by posts set in a rectangle"
Hence choice A is more appropriate and concise. But from a grammatical standpoint I don't think there's anything wrong with choice E. It's just that A is more appropriate as well as concise.