If we let  and  be the sidelengths of  and , respectively; the statements can be rewritten as:

Statement 1: 

Statement 2: 

Since the perimeter of an equilateral triangle is three times its common sidelength, comparison of the lengths of the sides is all that is necessary to determine which triangle, has the greater perimeter. The question can therefore be reduced to asking which of  and , if either, is greater.

Statement 1 alone is not sufficient to yield an answer:

Case 1: 

Case 2: 

Both cases satisfy Statement 1, but in the first case, , meaning that  has greater sidelength and perimeter than , and in the second case, , meaning the reverse. By a similar argument, Statement 2 is insufficient.

Now assume both statements to be true. The two equations together comprise a system of equations:

Multiply the first equation by 3 and the second by , then add:

Now substitute back:

 has the greater sidelength and, consequently, the greater perimeter.

Assume Statement 1 alone. Since in ,  is the right angle,  is the hypotenuse, which is longer than either of the two legs -  and . Since, in equilateral triangle , , it follows that  and . Consequently

,

and  has the greater perimeter.

Statement 2 alone gives insufficient information, since we know nothing about the second leg or the hypotenuse of . We see this by examining these two cases.

Case 1: Let  be an equilateral triangle of sidelength 6, and  in . If second leg  of  has length 4, then the triangle  is a 3-4-5 triangle, with hypotenuse  having length 5. The perimeter of  is , and the perimeter of  is .  has the greater perimeter.

Case 2: Let  be an equilateral triangle of sidelength 6, and  in . If second leg  of  has length 20,  has the greater perimeter on the basis of one side alone.

Recall the formula for perimeter of a triangle.

where  represents the side length of the triangle

Statement 1: We're given the length of the side so all we need to do is plug this value into the equation. 

Statement 2: We're given the area so we first need to solve for the side length.

Now we can plug the value into the equation, like we did in Statement 1.

To prove triangle congruence, we need to establish some conditions involving side and angle congruence.

We know  from reflexivity.

If we only assume Statement 1, then we know that , both angles being right angles. If we only assume Statement 2, then we know that  by definition of a bisector. Either way, we only have one angle congruence and one side congruence, not enough to establish congruence between triangles. The two statements together, however, set up the Angle-Side-Angle condition, which does prove that .

To prove triangle congruence, we need to establish some conditions involving side and angle congruence.

We know  from reflexivity.

From Statement 1 alone, by definition,  and, since both  and  are right angles, . This sets the conditions to apply the Side-Angle-Side Postulate to prove that .

From Statement 2 alone,we know that  by definition of a bisector. But we only have one angle congruence and one side congruence, not enough to establish congruence between triangles. 

Assume Statement 1 alone. If , then  and . That  does not prove or disprove the congruence statement. Therefore, Statement 1 alone - and by a similar argument, Statement 2 alone - is not sufficient to answer the question.

Now assume both statements to be true.

Suppose . Then this, along with the two statements, can be combined to yield the statement

.

Similarly, if ,  

,

and .

 and  cannot both be true, so it is impossible for .

From Statement 1 alone, we are only give one angle congruency and one side congruency, which, without further information, is not enough to prove or to disprove triangle congruence. By a similar argument, Statement 2 alone is not sufficient either.

Assume both statements are true. By Statement 1, the hypotenuses are congruent, and by Statement 2, one pair of corresponding legs are congruent. These are the conditions of the Hypotenuse Leg Theorem, so  is proved to be true.

Each statement alone give information about only one of the triangles; without information about the other triangle, it is impossible to prove or to disprove triangle congruence.

Assume both statements are true. For , it must hold that both  and . If , then since , then , and since , then . Therefore, . Similarly, if , then , and ; again, . Therefore,  the two statements together prove that  is false.

Assume both statements to be true. 

 and , so hypotenuse  can be calculated using the Pythagorean Theorem:

This establishes that  - that is, that the hypotenuses of triangles are congruent. This gives us one side congruence and one angle congruence, the right angles, between the triangles; however, we are not given any other side or angle congruences, so we cannot determine whether or not the triangles are congruent.

Assume Statement 1 only. By the Pythagorean Theorem, , so ; subsequently, , and one side congruence is proved. However, this, along with one angle congruence - the congruence of right angles  and  - is not enough to prove or disprove triangle congruence.

Assume Statement 2 only. By the Pythagorean Theorem, , so ; subsequently, , and one side congruence is proved. For the same reason as with Statement 1, this provides insufficient information.

The two statements together, however, are sufficient. Statements 1 and 2 estabish congruence between the hypotenuses and corresponding legs, respectively, setting up the conditions of the Hypotenuse-Leg Theorem. As a consequence, .