Assume Statement 1 alone. The ratio of the perimeters does not in and of itself establish similarity, since only one angle congruence is known.

Assume Statement 2 alone. The equation can be rewritten as a proportion statement:

This establishes that two pairs of corresponding sides are in proportion. Their included angles are both right angles, so , and  follows from the Side-Angle-Side Similarity Theorem.

Assume Statement 1 alone. The statement used to find a sidelength ratio:

However, since we only know one sidelength ratio, similarity cannot be proved or disproved.

From Statement 2, another ratio can be found:

Again, since only one sidelength ratio is known, similarty can be neither proved nor disproved.

Assume both statements to be true. Similarity, by definition, requires that 

From the two statements together, it can be seen that , so .

The acute angles of a right triangle are complementary, so  and  are a complementary pair, as are  and .

If Statement 1 is assumed—that is, if  and  are a complementary pair—then, since two angles complementary to the same angle—here, —must be congruent, . Since right angles ,   follows by way of the Angle-Angle Similarity Postulate, and Statement 1 turns out to provide sufficient information. By a similar argument, Statement 2 is also sufficient.

Assume Statement 1 alone is true. Then, since , both being right angles, and  from Statement 1,  follows by way of the Angle-Angle Similarity Postulate. A similar argument shows Statement 2 also provides sufficient information.

Assume both statements are true.

While in two similar triangles, the ratio of the perimeters, given in Statement 1, is indeed equal to that of the ratios of the lengths of the hypotenuses, given in Statement 2, this is not a sufficient condition for similarity. For example:

Case 1: 

Case 2:

In each case, the conditions of the main problem and both statements are met, since:

Both triangles are right - each Pythagorean triple is a multiple of Pythagorean triple 3-4-5;

The ratio of the perimeters is ; and,

.

But in Case 1, 

, since , and the similarity follows by way of the Side-Side-Side Similarity Principle. 

In Case 2, 

, since . This violates the conditions of similarity (note that in both cases, , but this is a different statement).

The two statements together are inconclusive.

Each of Statement 1 and Statement 2 gives information about only one of the triangles, so neither statement alone is sufficient.

Assume both statements are true. From Statement 1,  and  is right and measures .

From Statement 2 alone,  is isosceles; the acute angles of an isosceles right triangle must both measure , so, in particular,  . Also, it is given that  is right.

 and  (both of the latter being right angles), and by the Angle-Angle Postulate, .

Refer to the figure below, which gives the two rectangles with their diagonals as described.

Assume Statement 1 alone. The diagonals of a rectangle bisect the rectangle into congruent triangles, so  and . Congruent triangles are also similar, so it follows that  and . Since, by Statement 1,  - or, stated differently,  - by transitivity of similarity, 

, and

.

Assume Statement 2 alone. The quadrilaterals are rectangles, so , both being right angles. From Statement 2, , setting up the conditions of the Angle-Angle Postulate; therefore, .

Assume both statements are true. We show the two statements together are insufficent to prove or disprove that 

Suppose 

.

Then Rectangle  has perimeter 

and area 

Now set up two cases with different dimensions for Rectangle .

Case 1:

The perimeter of Rectangle  is 

,

one-third that of Rectangle .

The area of Rectangle  is

, 

one-ninth that of Rectangle .

,

so by the Side-Angle-Side Similarity Theorem, .

Case 2:

The perimeter of Rectangle  is 

,

one-third that of Rectangle .

The area of Rectangle  is

, 

one-ninth that of Rectangle .

Sides are not in proportion, making the statement  false.

Assume both statements are true. Consider these two scenarios:

Let 

and 

Since  and , then by the Hypotenuse-Leg Theorem, , and it follows that . 

Now, let 

and .

The sides are not in proportion, as , so .

In both scenarios, since , the triangles are right by the converse of the Pythagorean Theorem, satisfying the main premise; the triangles are scalene, having sides of three different lengths, satisfying Statement 1; and the triangles both have perimeter , satisfying Statement 2. The two statements together are insufficient.

Assume Statement 1 alone. By the 45-45-90 Theorem, an isosceles right triangle has two acute angles that each measure ; specifically,  and . By the Angle-Angle Similarity Postulate, .

Assume Statement 2 alone. The equality of the perimeters of two right triangles does not imply their similarity or nonsimilarity. For example:

Let 

and 

Since  and , it follows by the Hypotenuse-Leg Theorem that , and, since congruent triangles are similar, it further follows that . 

Now, let 

and .

The sides are not in proportion, as , so .

In both scenarios, since , by the converse of the Pythagorean Theorem, the triangles are right, and the triangles both have perimeter , thereby satisfying the main premise and Statement 2. Statement 2 alone is therefore insufficient.