Poor at proof questions could someone help me out?

1.

The angle AB ^ C is the angle at the circumference whose center angle AO ^ C is a flat angle which means that AB ^ C is a right angle.

2. 

We observe that the segments that join the center O to the vertices of the quadrilateral are congruent with each other.

AO ≡ BO ≡ CO ≡ DO.

If in a quadrilateral the diagonals bisect each other then the quadrilateral is a rectangle.

So ABCD is a rectangle.

There is insufficient information. If AC  is a diameter (ie angle AOC is 180) then angle ABC is a right angle - proving it depends on what you are allowed to assume and ultimately the angle in a semicircle is just known to be a  right-angle. However, even tho AOC looks like 180 on the diagram, there is no geometric reason that is has to be.