a) a = 3x, b = 21, c = x^2, and d = 7x
b) x^2 + 10x + 21
c) see below!
what you want to do here is use the formula you know for the area of a rectangle, length * width. start with rectangle a. the length of this one is x, the width is 3, so the area must be 3x. for b, the area must be 7*3, or 21. for c, the area must be x*x, or x^2. for d, the area is 7*x, or 7x. if it , cover everything besides the rectangle you're focusing on so you don't get distracted by the unnecessary numbers.
so, from all that, you have
a = 3x, b = 21, c = x^2, and d = 7x.
next, you want to use these to write a polynomial for the area of the large rectangle. well, we just found the area for all the rectangles inside this big rectangle, so all you have to do now is add up those smaller areas to get the overall area = x^2 + 7x + 3x + 21 = x^2 + 10x + 21. (a "simplified" polynomial just means one that's arranged from highest degree/exponent to lowest degree/exponent. x^2 goes first.)
now, instead of breaking it into smaller rectangles, you could just cover everything in the middle and look at those 4 values on the outside. how would you use those to find the area? again, your area formula is length*width. for the top side, part of it is x, and the other part is 7. therefore, that side has a length of x + 7. for the left side, part of it is 3, and the other part is x, so it has a width of x + 3. remember length * width. so now all you do is take those two lengths like this:
area = length*width = (x + 3)(x + 7)
and multiply that out, if you're familiar with the foil method, and you should get x^2 + 10x + 21, which is identical to our answer to part b.
given: ad ≅ bc and ad ∥ bc
prove: abcd is a parallelogram.
1. ad ≅ bc; ad ∥ bc 1. given
2. ∠cad and ∠acb are alternate interior ∠s 2. definition of alternate interior angles
3. ∠cad ≅ ∠acb 3. alternate interior angles are congruent
4. ac ≅ ac 4. reflexive property
5. △cad ≅ △acb 5. sas congruency theorem
6. ab ≅ cd 6. corresponding parts of congruent triangles are congruent (cpctc)
7. abcd is a parallelogram 7. parallelogram side theorem