Statement 1:  The perimeter of the square is known.

A square has four equal sides.  Write the perimeter formula for squares.

The side length of a square is a fourth of the perimeter.

Statement 2:  A secondary square with a known area encloses the primary square where all 4 corners of the primary square touch each of secondary square's edges.

If the primary square corners touch each of the secondary square edges, they must touch at the midpoint of each edge.  Since Statement 2 mentions that the secondary square area is known, it is possible to solve for the edge length and the diagonal of the secondary square.  Write the formula for the area of a square.

The diagonal of the secondary square can be solved by using the Pythagorean Theorem.

The side length of the secondary square also must equal the diagonal of the primary square.

Therefore:

Each of the statements, 1 and 2, provide something integral to calculating the length of a diagonal: an angle and the length of the sides connected to its vertex. The diagonal would be the third leg of the resulting triangle, and can be calculated using the law of cosines:

Since this is a parellelogram, knowing one angle allows us to know all the angles.

Area alone is not enough information. Imagine, for instance, a parallelogram with a shorter side of  and a longer side of . The diagonal would be well above 

However, with the perimeter, the smaller and larger sides must add up to one half of it, .

The longer diagonal reaches its maximum with the larger internal angle widens towards  degrees, and the parallelogram flattens into a line. Using the law of cosines, this translates to:

.

At its max, the diagonal could be no greater than 

To find the area we need the length and width of the rectangle. We can use II together with I to make an equation for perimeter with only one unknown.

So we need both to solve.

Solve for  and then go back to find  and then with that you can find the area of the base and you are finished.

When asked to find the width of a rectangle we will need to use both statemests together.

For Statement I) we can use the perimeter formula.

Now, for Statement II) we will use the length of the diagonal along with the Pythagorean Theorem.

From here you can solve the perimeter equation in terms of either l or w. Then you can use substitution into the Pythagorean Theorem to solve for a possible width. 

Statement 1: 

Statement 1 is sufficient to answer the question

Statement 2: 

Statement 2 is also sufficient to answer the question

Statement 1:      

Recall the formula to find the perimeter of a rectangle. Substitute in the given information and solve.

Statement 2: 

Recall the formula for the area of a rectangle. Substitute in the given information and solve.

Each statement alone is sufficient to answer the question.

The length of one diagonal alone does not prove the parallelogram to be a rectangle, nor do the lengths of the sides.

Suppose we know all of these lengths, though. Since  is a parallelogram, if , then . 

The sides  and diagonal   form a triangle  with sidelengths 25, 60, and 65. The parallelogram is a rectangle if and only if  is a right angle; therefore, we must determine whether the conditions of the Pythagorean Theorem hold:

This is true;  is a right angle and  is a rectangle.

Therefore, both statements together are sufficient to answer the question, but neither statement alone is sufficient to answer the question.

In order to find the diagonal, we must know the sides of the rectangle or know whether the triangles ADC or ABD have special angles.

Statement 1 alone doesn't let us calculate the hypothenuse of the triangles, because we only know one side.

Statement 2 alone is sufficient because it allows us to find all angles of the triangles inside of the rectangle. We can see that they are special triangles with angles 30-60-90. Any triangle with these angles will have its sides in ratio , where  is a constant. Here, , knowing this, we can calculate the length of the hypothenuse, also the diagonal, which will be .

Hence, statement 2 is sufficient.

I) Gives us the length of ASOF's diagonal. This by itself does not give us any way of finding the other sides.

II) Gives us one side length. From there we can use the perimeter to find the other side length and then the area.

Therefore, Statement II is sufficient to answer the question.