Since we are told that triangle ABC is a right triangle, to find the height, we just need the length of at least 2 other sides. From there, we can find the length of the height since in a right triangle, the height divides the triangle into two triangles with the same proportions. In other words . Therefore, we need to know the length of the sides of the triangle.

The height of a right triangle will be one of its side lengths.

I) tells us the length of our hypotenuse.

II) gives us the other two angle measurements.

They are both 45 degrees, which makes JKL a 45/45/90 triangle with side length ratios of  .

Which we can use to find the height.

Statement 1: 

More information is required to answer the question because our base and height can be  and  or  and  

Statement 2: We're given the base so we can narrow down the information from Statement 1 to  and . If the base is , then the height must be . 

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Statement 1: 

Additional information is required because our base and height can be  and ,  and , or  and . 

Statement 2: 

Even if we solve for our two values, we will not be able to determine which is the base and which is the height.

Statements 1 and 2 are not sufficient, and additional data is needed to answer the question.

If Statement 1 is assumed, then by the converse of the Pythagorean Theorem, the triangle is a right triangle with right angle , which is explicitly stated in Statement 2. If  is a right angle, then the other two angles are acute, since a triangle must have at least two acute angles. A right angle measures  and an acute angle measures less, so from either statement, we can deduce that  is the angle with greatest measure.

A right triangle must have its two acute angles complementary; if Statement 1 is assumed, then this is false, the triangle is not a right triangle, and  is not a right angle.

If Statement 2 is assumed, then we apply the converse of the Pythagorean Theorem to show that the triangle is not right. The sides of a triangle have the relationship 

only in a right triangle. If , then the statement to be tested would be 

This statement is false, so the triangle is not a right triangle, and  is not a right angle. 

Since we are already told that triangle ABC is a right triangle, we just need to find information about other angles or other sides.

Statement 1 allows us to calculate , simply by using the sum of the angles of a triangle, since we know AEC is also a right triangle because AE is the height.

Statement 2 is also sufficient because it allows us to know angle . Indeed, in a right triangle, the height divides the triangles in two triangles with similar properties. Therefore angle  is the same as .

Therefore, each statement alone is sufficient.

In order to find the angles of right triangle ABC, we would need to find the length of the sides and maybe found that the triangle is isoceles, or is a special triangle with angles 30-60-90. 

Statement one tells us that the height is equal to half the hypothenuse of the triangle. From that we can see that the triangle is isoceles. Indeed, an isoceles right triangle will always have its height equal to half the length of the hypothenuse. Therefore we will know that both angles are 45 degrees. Statement 1 alone is sufficient.

Statement 2 alone is insufficient because we don't know anything about the other sides of the triangle. Therefore it doesn't help us.

For , it is necessary that corresponding angles be congruent - specifically,  and . We show that the statements together are insufficient by assuming them both to be true and examining two cases:

Case 1: .

Case 2:   and .

Both cases fit the main body of the problem and both statements, but in the first case, , and in the second case, . The statement  holds only in the first case but not in the second. (Note that in both cases, , but this is a different statement.)

Corresponding angles of similar triangles - including the same triangle, considered in different configurations - are congruent. It follows from Statement 1 alone that . The measures of the acute angles of a right triangle add up to , so:

Assume Statement 2 alone. By the 45-45-90 Theorem, since the legs of the right triangle are of equal length, the acute angles of the triangle, one of which is , measures .