In the diagram above, O is the center of the circle and ACDE is a squa

Why can we assume that AB is the diameter? In theory, couldn't Angle AOB be something other than 90 degrees and we need statement 2 to guarantee that AB is the diameter?

I think you meant 180 degrees instead of 90 degrees. Anyway, check the highlighted parts below:

Problem Solving
Figures: All figures accompanying problem solving questions are intended to provide information useful in solving the problems. Figures are drawn as accurately as possible. Exceptions will be clearly noted. Lines shown as straight are straight, and lines that appear jagged are also straight. The positions of points, angles, regions, etc., exist in the order shown, and angle measures are greater than zero. All figures lie in a plane unless otherwise indicated.

Data Sufficiency:
Figures:
• Figures conform to the information given in the question, but will not necessarily conform to the additional information given in statements (1) and (2).
• Lines shown as straight are straight, and lines that appear jagged are also straight.
• The positions of points, angles, regions, etc., exist in the order shown, and angle measures are greater than zero.
• All figures lie in a plane unless otherwise indicated.

Hope it helps.

The answer is A.

Stmt 1. It the radius is 2 then the Diameter AB is 6. In triangle ABC Angle ACB is a right angle. Using Pythagoras Theorem we can find out the length of AC and hence the area of square. Sufficient.

Stmt 2. Angle ACB is 90 degree. We already know this from the question stem so it does not provide any additional information and hence is insufficient.

Hey, I had a similar doubt as explained before but slightly more nuanced.

While yes the guidelines say that the line is straight which is mentioned to be the case for AB. The same guideline could be interpreted as OA and OB are the straight lines used. And the difference angle AOB is not 180 but rather 179.5+ which would still follow the guidelines and not be a diameter.

Furthermore, the diagram mentions a square with the vertices ACDE.
It does not mention :
- ABC is a triangle, thereby eliminating a quadrilateral of AOBC.
- Point is either on the circumference or inside or outside the circle, explicitly.
- Inscribed triangles
- OC is a radius.
Any of these would clearly imply that line AB is straight and as a result, ABC is an inscribed triangle and such.

BUT, the inclusion of the second statement, categorically proves ABC is an inscribed triangle - AB is diameter - points are on the line.
Thus making both statements sufficient - Ans choice C.

I do understand that once AB is assumed to be a single line the entire explanation adds up - no doubts.

But given the lack of any of these points mentioned above being a part of the data, how are we to assume so?
Alternatively, if we are to assume so, would we be right to assume that any figure drawn with a triangle that is implied to be inscribed within a semi-circle can in fact ONLY be a triangle, and in no case can it be a quadrilateral?