To prove two figures similar, we must prove that their corresponding angles are congruent, and that their corresponding sides are in proportion.

All four sides of a rhombus are congruent, so it easily follows that corresponding sides of two rhombuses are in proportion, regardless of whether they are similar or not; it is therefore necessary and sufficient to prove that coresponding angles are congruent. Also, since a rhombus is a parallelogram, opposite angles are congruent and consecutive angles are supplementary—that is, their angle measures total . Therefore, it is neccessary and sufficient to prove just one pair of corresponding angles congruent.

Assume Statement 1 alone.  and  are not corresponding angles, so their congruence does not prove the rhombuses are similar; however it does not prove they are not similar either; for example, two squares are both rhombuses with four right angles, so they are similar and fit this condition.

Assume Statement 2 alone.  and  are two angles of the same rhombus, so similarity cannot be proved or disproved without information about the other rhombus.

Now assume both statements.   and  are consecutive angles of Rhombus ; a rhombus being a parallelogram, the degree measures of the angles total . From Statement 2, they are congruent, so each measures . Since  from Statement 1,  also measures . Since each parallelogram has at least one right angle, each has four right angles. Therefore, corresponding angles are congruent, so the rhombuses are similar.

To prove two figures similar, we must prove that their corresponding angles are congruent and that their corresponding sides are in proportion.

We show that Statement 1 alone provides insufficient information by examining two cases.

Case 1: Trapezoid  Trapezoid , with  a  angle.

A pair of base angles of an isosceles trapezoid are congruent, so, since  measures , so does . The leg angles of a trapezoid have measures whose sum is , so  and  measure . By congruence of corresponding angles, , so  measures , and the conditions of Statement 1 are met. Furthermore, two congruent triangles are also similar, so Trapezoid  Trapezoid .

Case 2: Examine the figure below, in which . Note that these congruent upper bases have been superimposed upon each other:

The trapezoids are isosceles since their base angles are congruent; also, the conditions of Statement 1 are met. However, , but . The sides are not in proportion, so the trapezoids are not similar.

Assume Statement 2 alone. Two congruent trapezoids whose side lengths fit the conditions of the statement are also similar. But examine this diagram, in which , , and . Note that the congruent upper bases have been superimposed upon each other:

The conditions of Statement 2 are met, but , so the trapezoids are not similar.

Now assume both statements to be true. As stated before, it follows from Statement 1 that all corresponding angles are congruent.  and , so it follows from the Division Property of Equality that

It remains to be demonstrated that  is equal to the above ratios as well. If the diagonals  and  are constructed,  and  are formed; since  and , by the Side-Angle-Side Similarity Theorem, . It follows that , and by angle addition, since , it follows that . From the Angle-Angle Postulate, the other two triangles formed are similar—that is, —and it follows that . Therefore, all four sides of the trapezoids are in proportion, and the trapezoids are similar.

To prove two figures similar, we must prove that their corresponding angles are congruent and that their corresponding sides are in proportion.

Assume both statements are true. We show that they provide insufficient information by examining two scenarios.

Case 1: Trapezoid  Trapezoid , then  and , so the conditions of both statements are met; also, since the trapezoids are congruent, they are also similar.

Now examine isosceles trapezoid  below, in which , , , and  are positioned on the bases so that  and .

Since  and , Quadrilateral  is a parallelogram, and ; similarly, . Therefore, Trapezoid  is also isosceles. Also, by the Corresponding Angles Theorem,  and , and the conditions of both statements are met. However, corresponding sides are not in proportion, since , but ; consequently, the trapezoids are not similar.

Similar rectangles (or any shape for that matter) are the same shape but can be different sizes. What that means is that their side lengths all follow a common ratio; if one pair of corresponding sides follow the ratio 2:1, then all corresponding sides must follow the same ratio.

Statement I gives us the perimeter and one side of .

Statement II gives us the area and one side of .

We can find all the sides of both rectangles, but we need both statements to do so. Once we have all side lengths, we can compare them to see if they follow the same ratios.

IF  has perimeter of 16 and one side is 3, we can find the other side using the following:

If  has area of 44 and side of 6, the other side can be found via the following:

Compare the ratios of the sides to find out whether the two rectangles are similar:

Therefore, the rectangles are not similar.

Statement 1:  The area is 1.

Given the area of a square is sufficient to find the diagonal of the square.  The formula to find the side length of the square is:

The Pythagorean theorem can then be used to solve for the length of the diagonal.

Statement 2:  The square is inscribed inside of a circle with all 4 corners touching the edge of the circle, and the circle has an area of 1.

This statement is also sufficient to solve for the diagonal.  With all 4 corners of the square touching the circle, the diagonal of the square is simply the length from one of the square's opposing 90 degree angles to its opposite angle, which is the diameter of the circle.  

Given the area of the circle, use the formula  to determine the diameter of the circle, which is also the diagonal of the square.

Therefore:

Find the length of the diagonal of rectangle DEFT

I) Side DE is 3 times the length of side TD

II) The perimeter of the rectangle is 

We need to find the diagonal of a rectangle. If we work back from there, we can see that to find the diagonal, we will need to the lengths of both sides of the rectangle.

I) Gives us the relationship between the two sets of sides:

II) Tells us the perimeter, for which we can write and equation:

Now, use the clue from I) to simplify our equation in II) to one variable:

So now that we have our side lengths, use Pythagorean Theorem to find the diagonal:

With a ratio of sides of 2:3, statement 1 tells us PS and QR are 9 units long. Knowing the ratio we can determine both PQ and SR are 6. Since rectangles have four 90 degree angles, we can use the Pythagorean Theorem to solve for the length of SQ: . Therefore, statement 1 alone is sufficient.

With a ratio of sides of 2:3, statement 2 tells us PQ and SR are 6 units long. Knowing the ratio we can determine both PS and QR are 9 units long. Since rectangles have four 90 degree angles, we can use the Pythagorean Theorem to solve for the length of SQ:  . Therefore, statement 2 alone is sufficient.

Therefore, the correct answer is  EACH statement ALONE is sufficient.

By definition, a parallelogram is a quadrilateral with 2 sets of parallel sides and diagonals that bisect each other.

From statement 1 we cannot determine the length of AC, since ABCD is only defined as a parallelogram and not necessarily a rectangle (which is a special parallelogram with equal diagonals). Therefore, statement 1 alone is not sufficient.

From statement 2, we can determine that  since the definition of a parallelogram states the diagonals bisect each other. Therefore, Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

With the information from statement 1, we don't know the measurements of any parts of PQRS. Therefore,  Statement 1 alone is not sufficient.

With the information from statement 2, we know that  , but since we have no confirmed definition of the shape of PQRS, we can't extrapolate the length of SQ to any other length.  Therefore, statement 2 alone is not sufficient.

Even if we look at both statements 1 and 2 together, quadrilateral PQRS looks like a rectangle, but we are only told it is a parallelogram. The figures on the GMAT are not drawn to scale, therefore your eyes cannot be trusted. No difinitive length of PR can be determined. Therefore, statements 1 and 2 together are not sufficient.

Therefore, the correct answer is Statements (1) and (2) TOGETHER are NOT sufficient.

To find the diagonal of a rectangle, we need the length of two sides.

I) Gives us the length of one side.

II) Lets us find the length of the next side.

Use I) and II) with Pythagorean Theorem to find the diagonal. 

Conversely, recognize that we are making a 3/4/5 Pythagorean Triple and see that the last side is 30 fathoms.