o.k. I'll give you "some" help, but you can do the work yourself.
1. you have a quadrilateral with certain vertices. You wish to prove it is an isosceles trapazoid. What do you know about an isosceles trapazoid???
Well, you know that:
1. the bases are parallel
2. the sides are equal
3. the "base" is longer than the "top"
What do you have to work with?? only the co-ordinate points.
Well, it 'pears to ME that you can use the distance formula to find the length of the sides and show that they are equal. You can then find the slope of the base and top to show they are parallel. Then you can find the length of the base and top to show they are UNEQUAL and thus you do not have a parallelogram.
problem #2. What are the characteristics of a square. Well, if I recall correctly, it is 4 sides equal and one right angle. Thus, you can use the distance formula to show all the sides are equal. THEN, you can use the distance formula to show one diagonal is the sum of the squares of two sides, thus proving the included angle is a right angle.
Don't make this stuff harder than it is.