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.

Both rectangles, by definition, have four right angles.

A rhombus is a quadrilateral with four sides of equal length. A square is a quadrilateral with four right angles and four sides of equal length. Therefore, the two statements are actually equivalent in this context, so either they together provide insufficient information, or either alone does. We show that the latter is the case.

If both statements are true, and the rectangles are rhombuses and squares, then

 and .

Consequently,  

.

,

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

Assume Statement 1 alone. The area of the circle that circumscribes  is four times the that of the circle that circumscribes , so the radius of the former is the square root of this, or two, times the radius of the latter; therefore, the ratio of the diameters of the circles is also two. Since in a right triangle inscribed inside a circle, the hypotenuse must be a diameter of that circle, 

 or 

However, no other information is given or can be determined about any other sides or angles, so the triangles cannot been proved or disproved to be similar.

Assume Statement 2 alone. , or , but again, no other information is given or can be determined about any other sides or angles, so the triangles have not been proved or disproved to be similar.

Now assume both statements. , setting up the conditions for the Hypotenuse-Leg Similarity Theorem; it follows that .

Assume Statement 1 alone. If , then corresponding angles are congruent - specifically, . Statement 1 directly contradicts this, so .

Statement 2 alone does not prove or disprove similarity, since information is only given about one set of corresponding sides. No information is given about corresponding angles, or any other set of corresponding sides. 

Similarity of two triangles does not require that corresponding sides be congruent - only that they be in proportion. Therefore, the inequality of one pair of corresponding sides, as given in Statement 1 alone, is insufficient to prove or disprove that .

However, similarity of two triangles requires that all three pairs of corresponding angles be congruent. Therefore, Statement 2 alone, which establishes that one such pair is noncongruent, proves that .

Assume Statement 1 alone. Rectangle   is not a rhombus, so  or  . Rectangle  is a rhombus, so . Therefore, either 

or 

Either way, the corresponding sides are not in proportion; it follows that .

Assume Statement 2 alone. No information is given about  , so the quantities  and  cannot be proved to be equal or unequal. If they are unequal, since the sides are not in proportion, it follows that . If they are equal, then, since , it follows by the Side-Angle-Side similarity Theorem that .

Assume Statement 1 alone. Since Quadrilateral  is a trapezoid, it follows that ; by the Corresponding Angles Principle, . By reflexivity, . By the Angle-Angle Postulate, .

Assume Statement 2 alone.

Also, by reflexivity, . By the Side-Angle-Side Similarity Theorem, it follows that .

Assume Statement 1 alone.  and  have the same degree measure, so the inscribed angles that intercept these arcs must also have the same degree measure - that is, . Since , both being right angles, this sets up the conditions of the Angle-Angle Postulate, so it follows that .

Assume Statement 2 alone. Major arc  and major arc .   by Statement 2, so 

Again, the inscribed angles that intercept these arcs must also be congruent - that is, . Again, this, along with , prove that  by way of the Angle-Angle Postulate. 

To know the length of the height triangle, we would need to know the lengths of the triangle or the angles to have more information about the triangle.

Statement 1 only gives us a length of a side. There is nothing more we can calculate from what we know so far.

Statement 2 alone tells us that the triangle is isoceles. Indeed, ABC is a right triangle, if one of its angle is 45 degrees, than so must be another. Now, we are able to tell that the length of the height would be the same as half the hypothenuse. A single side would be sufficient to answer the problem. Statment 1 gives us that information. Therefore, both statements together are sufficient.