Julia,

Sometimes with these coordinate geometry problems, its helpful to draw out the polygon that you are working with.

Fist remember that the letters of the vertices must always follow the same order as given in the problem. So your polygon will look something like this:

T=(8a+33) __________________ U=(10a-66)

W=(7b-59) __________________ V=(2b+41) (imagine the sides drawn in ;) )

Since you have the values of a and b it would be best to plug those in first so you can determine the coordinates of all of the vertices.

Now, the easiest way to prove a parallelogram is to prove both pairs of opposites are equal lengths. To do this simply apply the distance formula.

(here's a website if youneed help with that: http://www.cliffsnotes.com/study_guide/Distance-Formula.topicArticleId-18851,articleId-18841.html )

Compare the distance between T and U with the distance between W and V, if these are equal than the sides are parallel.

Then compare the distance between T and W with U and V; if these are equal then the bases are parallel...Resulting in a parallelogram by definition. If either of these pairs are not equal, your polygon is just a quadrilateral.