We show that both statements together give insufficient information. Assume both statements together. Consider the following two cases:

Case 1: 

Since the angles of measures  and  form linear pairs with the angle of measure , each is supplementary to that angle and, subsequently,  .

The angle of measure  is vertical to the angle of measure , so the two must be congruent; .

From Statement 1, 

Since  and , then .

These values are therefore consistent with the diagram and with both statements.

Case 2: 

We can find the values of the other variables as before:

, so .

Again, all values are consistent with the diagram and both statements. 

Since at least two different values of  satify the conditions, the two statements are insufficient.

Statement 1 alone gives us that , but reveals no clues about any of the eight angle measures. From Statement 2 alone, that , we can assume that , and  all have measure , but no clues are given about any of the other four angles—in particular, .

Assume both statements are true. From Statement 1, , and by way of the Parallel Postulate, corresponding angles have the same measure—in particular, . From Statement 2, we know that . From these two statements, .

If two chords of a circle intersect inside it, then the measure of an angle formed is equal to the arithmetic mean of the measures of arc it intercepts and the arc its vertical angle intercepts. In other words, in this diagram,

Both arc measures are needed to find the measure of the angle. Neither statement alone give both; the two together do.

We can use the side length and the Pythagorean Theorem to find the diagonal of a square.

We can find side length from area, so we could solve this with either I or II.

The diagonal of the square can be calculated as long as we have any information about the lengths or area of the circle or of the square.

Statement 1, by giving us the area of the circle, allows us to find the radius of the circle, which is half the length of the side. Therefore statement 1 alone is sufficient.

Statement 2, by telling us the length of a side of the square is also sufficient, and would allow us to calculate the length of the diagonal.

Therefore, each statement alone is sufficient.

We are asked to find the length of a diagonal of a square.

We can do this if we have the side length. We can find side length from either perimeter or area.

From Statement I)

In this case, our side length is 15 meters.

We can use this and Pythagorean Theorem or 45/45/90 triangles to find our diagonal.

From Statement II)

From here, we can plug the side length into the Pythagorean Theorem like before and solve for the diagonal.

Therefore, either statement alone is sufficient to answer the question.

Statement 1: The information provided would only be useful if the ratio of square A to square B was known. 

Statement 2: We need the length of the square's side to find the length of the diagonal and we can use the area to solve for the length of the side. 

Now we can find the diagonal: 

The length of the diagonal of a square is given by , where  represents the square's side. As such, we need the length of the square's side.

Statement 1: 

Statement 2: 

Both statements provide us with the length of the square's side. 

 To solve this problem, we must recognize that the diagonal bisector creates identical 45˚ - 45˚ - 90˚ right triangles. This means that, if the sides of the square are  then the diagonal must be . We can then set up the following equation:

If  then the area must be:

A rectangle, by definition, is a parallelogram. Statement 1 asserts that the diagonals of this parallelogram are perpendicular. Statement 2 asserts that adjacent sides of the parallelogram are congruent, so, since opposite sides are also congruent, this makes all four sides congruent. From either statement alone, it can be deduced that Rectangle  is a rhombus. A figure that is a rectangle and a rhombus is by definition a square.