From Statement 1 alone, the circumscribed circle has as its diameter the circumference  divided by , or 20. From Statement 2 alone, the circle has area , so its radius can be found using the area formula;

The diameter is the radius doubled, which here is 20.

The hypotenuse of a right triangle is a diameter of the circle that circumscribes it, so the diameter of the circle gives us the length of the hypotenuse. However, we are looking for the length of a leg, . Either statement alone gives us only the length of the hypotenuse, which, without other information, does not give us any further information about the right triangle.

Assume Statement 1 alone. The hypotenuse of a right triangle is longer than either of its two other sides; since  and   are equal in length, neither is the hypotenuse. This leaves  as the hypotenuse.

Assume Statement 2 alone. Since , neither can be the right angle. Therefore,  is the right angle, and its opposite side, , is the hypotenuse.

Statement 1 alone gives insufficient information, as it only establishes congruence between two pairs of corresponding sides; without a third congruence or noncongruence between sides or angles, it cannot be established whether the triangles themselves are congruent. Statement 2 alone is insufficient, since it only compares two angles without giving any information about sidelengths, whether absolute or relative.

Assume both statements to be true. From Statement 1, we have the congruence statements  and , and from Statement 2, we have that the included angle  from  has measure , and that the included angle  from , being obtuse, has, by definition, measure greater than this. This sets up the conditions of the Side-Angle-Side Inequality Theorem, or Hinge Theorem, which states that in this situation, the third side opposite the greater angle is longer than the third side opposite the lesser. Therefore, it can be deduced that .

Either statement alone is sufficient.

From either statement alone, it can be determined that  and ; each statement gives one angle measure, and the other can be calculated by subtracting the first from , since the acute angles of a right triangle are complementary. 

Also, since  is the right angle,  is the hypotenuse, and , opposite the  angle, the shorter leg of a 30-60-90 triangle. From either statement alone, the 30-60-90 Theorem can be used to find the length of longer leg . From Statement 1 alone,  has length  times that of the hypotenuse, or . From Statement 2 alone,  has length  of the shorter leg, or .

This is a Pythagorean theorem question.  The lengths of a right triangle are related by the following equation:    In the problem statement,  and  Therefore, 

All right angles are congruent, so .

Since Statement 1 tells us that , this sets up the conditions for the Angle-Angle Similarity Postulate, so .

Statement 2 alone only tells us their hypotenuses. Congruence between one pair of angles and the measures of one pair of sides is insufficient information to determine whether two triangles are similar (given one angle, at least two pairs of proportional sides are required).

Therefore, the answer is that Statement 1 alone, but not Statement 2, is sufficient.

Each statement alone only gives a relationship between two sides within one triangle, so neither alone answers the question of the similarity of the two triangles.

Assume both statements are true. Then, since ,  . 

By the multiplication property of inequality, since 

 and ,

Since, by definition,  requires that , .

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.