Sentence Correction

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

I am stuck between A and E. My gut tells me E. I believe "with" is correct.

OA is (A)....

Let's keep this in mind....' flat-topped dwellings known as shades'. This means that there are dwelling places known as shades and they are more than one.But we want to talk about their roofs.

Option A.'each a roof of poles and arrowweed '.This is an incorrect English because we cant have something like' each a roof.'

Option B. this is similar to option A same improper use of English. 'each a roof of poles and arrowweed'

Option C; This is the correct answer ' with each being a roof of poles and arrowweed' because it clearly illustrates the expression laid down in the first tense and it option is in line with the authors description of the dwellings known as shades.

Option D; 'with roofs of poles and arrowweed to be ' This option is incorrect because 'to be' illustrates what will happen later in the futures which contradicts the tenses usage of sentence.It was fully expressed in past tense and verb 'to be' will not fit in.

Option E ; 'with roofs of poles and arrowweed that are' this should have been a suitable answer but the use of the singular past tense verb 'are' makes the incorrect,because there were 'dwellings known as shades' and they are more than one.Thus,it suppose to attract past tense plural verb ,not singular