We are being asked to compare the lengths of the hypotenuses of the two triangles, since  and  are the sides opposite the right angles of their respective triangles.

Statement 1 alone gives insufficient information, as shown by examining these two cases.

Case 1: 

By the Pythagorean Theorem, the hypotenuse  has length

The hypotenuse  has length

Case 2: 

The hypotenuse  has length

and, as in Case 1,  has length .

In both cases,  and , so . But in the first case,  was longer than , and in the second case, the reverse was true.

Statement 2 is insufficient in that it only gives us the congruence of one set of corresponding legs; without further information, it is impossible to determine which hypotenuse is longer.

Now assume both statements are true. Since  and , by the subtraction property of inequality, 

and 

It follows from  and  that  and ; by the addition property of inequality, 

By the Pythagorean Theorem, 

and 

,

so the above inequality becomes, by substitution,

and 

,

proving that  is longer than .

In a right triangle, the angle of greatest measure is the right angle, and the side opposite it is the hypotenuse.

Assume both statements are true. We can eliminate  as the right angle, as it has measure less than both  and . However, we have no information that tells us which of   and  has the greater measure, so we cannot determine which is the right angle. Subsequently, we cannot eliminate either of their opposite sides,  or , respectively, as the hypotenuse.

The two statements together only give information about the angle measures of the two triangles. Without any information about the relative or absolute lengths of the sides, no comparison can be drawn between their hypotenuses.

Statement 1 alone gives insufficient information.  and , but it is not clear which of the three sides is the hypotenuse of .  is not the longest side, so we know that  or  is the hypotenuse, and the other is the second leg. We explore the two possibilities:

If  is the hypotenuse, then the legs are  and ; since the lengths of the legs are 12 and 24, by the Pythagorean Theorem,  has length

.

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 alone gives insufficient information in that it only gives information about the angles, not the sides.

Assume both statements are true. If  is the hypotenuse and  is a leg, then, since the hypotenuse measures twice the length of a leg from Statement 1, the triangle is 30-60-90, contradicting Statement 2. Therefore, by elimination,   is the hypotenuse, and, consequently, .

Since we are comparing angles, we need to identify the angle of greatest measure; in a right triangle, the angle of greatest measure is the right angle, and the side opposite it is the hypotenuse.

Statement 1 is insufficient, since we can eliminate only angle  as the right angle, and, subsequently, only  as the hypotenuse. Similarly, Statement 2 is insufficent, since we can eliminate only angle  as the right angle, and, subsequently, only  as the hypotenuse. But if we are given both statements, we can eliminate  and  as the hypotenuse, leaving  as the hypotenuse.

We are being asked to compare the lengths of the hypotenuses of the two triangles, since  and  are the sides opposite the right angles of their respective triangles.

Assume Statement 1 alone. We have that , , and , both being right angles, thereby establishing congruence between two pairs of sides and a pair of included angles. By the Side-Angle-Side Theorem, , and, consequently  and  have equal length.

Assume Statement 2 alone. We are only given information about the angle measures, but nothing about the lengths of the sides - actual lengths or comparisons. We can make no conclusions about which hypotenuse is longer.

Since  is given as the right angle of the triangle , we are being asked to evaluate the length of hypotenuse .

Statement 1 alone gives insufficient information. We note that the area of a right triangle is half the product of the lengths of its legs, and we examine two scenarios:

Case 1: 

The area is 

By the Pythagorean Theorem, hypotenuse  has length

Case 2: 

The area is 

By the Pythagorean Theorem, hypotenuse  has length

Both triangles have area 24 but the hypotenuses have different lengths.

Assume Statement 2 alone. A circle that circumscribes a right triangle has the hypotenuse of the triangle as one of its diameters, so the length of the hypotenuse is the diameter - or, twice the radius - of the circle. Since the area of the circumsctibed circle is , its radius can be determined using the area formula:

The diameter - and the length of hypotenuse  - is twice this, or 10.

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.