Assume Statement 1 alone. Since we do not know whether  is the hypotenuse or a leg of , we can show that  can take one of two different values.

Case 1: If  is the hypotenuse, then the legs are  and ; since their lengths are 10 and 20, by the Pythagorean Theorem,  has length

.

Case 2: If  is a leg, then the hypotenuse, being the longest side, is , and  is the other leg; by the Pythagorean Theorem,  has length

.

Statement 2 gives insufficient information, since it only clues us in to the measures of the angles, not the sides.

Now assume both statements. Since one of the angles of the right triangle has measure , the other has measure ; the triangle is a 30-60-90 triangle, and therefore, its hypotenuse has twice the length of its shorter leg. Since, from Statement 1, ,  is the hypotenuse, and  is the longer leg, the length of which, by the 30-60-90 Theorem, is  times that of shorter leg , or .

As each statement alone only gives the length of one segment, neither statement alone is sufficient to find the length of any other segment, including, in particular, .

Assume both statements to be true.  can be found by way of the Pythagorean Theorem:

Now note that  is the altitude of the right triangle from the vertex of the right angle, which divides the right triangle into two triangles similar to each other and the large triangle. Specifically,

By similarity, 

.

The hypotenuse of a right triangle is its longest side. From Statement 1 alone, we can eliminate only   as the hypotenuse, and from Statement 2 alone, we can eliminate only  . From Statements 1 and 2 together, we can eliminate both, leaving  as the hypotenuse.

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 , .