In order for two triangles to be congruent, they must be identical. That is, the lengths of the corresponding sides of two congruent triangles must be equal. This means that in order for a triangle to be congruent to one with a height of    and a base of  ,  its hypotenuse must be the same length as the hypotenuse of that triangle, which we can find using the Pythagorean Theorem:

In order for two right triangles to be congruent, the bases and heights must have identical lengths. Since we have a given right triangle with a base of , the congruent right triangle must also have a base of  .

In order for two right triangles to be congruent, the hypotenuses and acute angles must be identical. Since we have a given right triangle with an acute angle of , the congruent right triangle must also have an acute angle of  .

In order for two right triangles to be congruent, the bases and heights must have identical lengths. Since we have a given right triangle with a base of , the congruent right triangle must also have a base of  .

The hypotenuse of the large right triangle is 

The area of the large right triangle is half the product of its base and its height. The base can be any side of the triangle; the height would be the length of the altitude, which is the perpendicular segment from the opposite vertex to that base.

Therefore, the area of the triangle can be calculated as half the product of the legs:

Or half the product of the hypotenuse and the length  of the dashed line.

To calculate , we can set these expressions equal to each other:

Using the formula for the area of a right triangle, we can plug in the given values and solve for the height of the triangle:

The height AE, divides the triangle ABC, in two triangles AEC and AEB with same proportions as the original triangle ABC, this property holds true for any right triangle.

In other words, .

Therefore, we can calculate, the length of AE: 

.

As we have previously seen, the height of a right triangle divides a it into two similar triangles with sides of same proportion.

Therefore, we can set up the following equality:  or .

By plugging in the numbers, we get,  or .

Since here ABC is a isosceles right triangle, its height is half the size of the hypotenuse.

We just need to apply the Pythagorean Theorem to get the length of BC, and divide this length by two.

 , so .

Therefore,  and the final answer is  .

Let one of the angles measure ; then the other angle measures . The sum of the measures of the acute angles of a triangle is , so we can set up and solve this equation:

The acute angles measure ; since we want the smaller of the two,  is the correct choice.