Statement 1 is insufficient, as it only gives that two sides are of equal length; it gives no side lengths, nor does it give any measurements that yield the side lengths. 

Assume Statement 2 alone. By the Triangle Inequality, the length of each side must be less than the sum of the lengths of the other two, so

Also, 

Therefore, 

,

and we can find the range of the values of the perimeter 

by adding:

Therefore, the perimeter may or may not be greater than 50.

Assume both statements to be true. An isosceles triangle has two sides of the same length, so either  or . 

If , the perimeter is 

If , the perimeter is 

Either way, the perimeter is less than 50.

For the sake of simplicity, we will assume the length of each side of the square is 1; this reasoning works independently of the side length. The perimeter of the square is, as a result, 4, and the length of each of the diagonals  and  is  times the length of a side, or simply .

The equivalent question becomes whether the perimeter of the triangle is greater than, equal to, or less than 4. The statements can be rewritten as

Statement 1: 

Statement 2: 

We show that these two statements together provide insufficient information. 

By the Triangle Inequality, the sum of the lengths of two sides of a triangle must exceed the third. We can get the range of values of  using this fact:

Also, 

So, 

Add  and  to all three expressions; the expression in the middle is the perimeter of :

Since , for all practical purposes, 

Therefore, we cannot tell whether the perimeter is less than, equal to, or greater than 4. Equivalently, we cannot determine whether the triangle or the square has the greater perimeter.

We demonstrate that both statements together provide insufficient information by examining two cases:

Case 1: 

The perimeter of  is

.

Case 2: .

The perimeter of  is

.

Both cases satisfy the conditions of both statements but different perimeters are yielded. 

Statement 1 is insufficient, as it only gives that two sides are of equal length; it gives no side lengths, nor does it give any measurements that yield the side lengths. 

Assume Statement 2 alone. By the Triangle Inequality, the length of each side must be less than the sum of the lengths of the other two, so

We can find the minimum of the perimeter:

:

The perimeter of  is greater than 50.

Assume Statement 1 alone. We show that this provides insufficient information by examining two scenarios.

Case 1: . By definition,  and , satisfying the condtions of the main body of the problem, and , satisfying the condition of Statement 1. Since the triangles are congruent, all three pairs of corresponding sides have the same length, so the perimeters are equal.

Case 2: Examine this diagram, which superimposes the triangles such that  and  coincide with  and , respectively:

The conditions of the main body and Statement 1 are met, since ,  (their being the same segment and angle, respectively, in the diagram), and  by construction. Note, however, that , so:

.

Making the perimeters different. 

Now assume Statement 2 alone.  is the included side of  and , and   is the included side of  and . The three congruence statements given in the main body and Statement 2 together set the conditions of the Angle-Side-Angle Postulate, so , and the perimeters are indeed the same.

Assume Statement 1 alone. A segment that has as its endpoints the midpoints of two sides of a triangle is a midsegment of the triangle, and its length is half that of the side to which it is parallel. Therefore, the sum of the lengths of the midsegments—that is, the perimeter of the triangle they form, which from Statement 1 is 34, is half the perimeter of the larger triangle, which here is . The perimeter of  is therefore 68.

Assume Statement 2 alone. Here,  itself is the triangle formed by the midsegments. Since the larger triangle has perimeter 136,  has perimeter half this, or 68.

Assume both statements to be true. An isosceles triangle has two sides of equal length, so either  or . By the Triangle Inequality Theorem, the length of each side must be less than the sum of the lengths of the other two; both scenarios are possible, since

and 

.

If , the perimeter of  is

.

If , the perimeter of  is

.

Without further information, it is impossible to determine whether the perimeter is less than or greater than 60.

From Statement 1 alone, we know that two sides are of length 10, but we are not given any clue as to the length of the third. Therefore, we cannot calculate the sum of the side lengths, which is the perimeter. Statement 2 alone is unhelpful, since it only gives that two angles have equal measure; applying the Converse of the Isosceles Triangle Theorem, it can be determined that their opposite sides  and  are congruent, but no actual side lengths can be found.

Now assume both statements to be true. From Statement 1, , and, as stated before, it follows from Statement 2 that . Therefore, the triangle is equilateral with sides of common length 10, making the perimeter 30.

If only one of the statements is known to be true, the only congruent pairs that are known between the triangles comprise two sides and a non-included angle; this information cannot prove congruence between the triangles. If both are known to be true, however, they, along with either of the given side congruences, set up the conditions for the Angle-Angle-Side Theorem, and the triangles can be proved congruent.

The answer is that both statements together are sufficient to answer the question, but not either alone.

We are given two triangles with two side congruences between them. If we compare their included angles (the angles that they form), the angle that is of greater measure will have the longer side opposite it. This is known as the Hinge Theorem.

The first statement says explicitly that the first included angle, , has greater measure than the second, , so the side opposite , , has greater measure than .

The second statement is not so explicit. But if  is a right angle,  must be acute, and if  is right, then , which again proves that .

The answer is that either statement alone is sufficient to answer the question.