To solve, simply use the Pythagorean Theorem.

Recall that an isosceles right triangle has two leg lengths that are equal.

Therefore, to solve for the hypotenuse let  and  in the Pythagorean Theorem.

Thus,

Based on the description of your triangle, you can draw the following figure:

You can do this because you know:

1. The two equivalent sides are given.
2. Since a triangle is  degrees, you have only  or  degrees left for the two angles of equal size. Therefore, those two angles must be  degrees and  degrees.

Now, based on the properties of an isosceles triangle, you can draw the following as well:

Based on your standard reference  triangle, you know:

Therefore,  is .

This means that  is  and the total base of the triangle is .

Now, the area of the triangle is:

 or 

Use the formula for area of a triangle:

1. Find the height of the triangle:

2. Use the formula for area of a triangle:

Based on the description of this question, you can draw your triangle as such. We can do this thanks to the nature of an isosceles triangle:

Now, you know that the area of a triangle is defined as:

So, for our data, we can say:

Solving for , we get:

Thus, .

Now, for our little triangle on the right, we can draw:

Using the Pythagorean Theorem, we know that the other side is:

This can be simplified to:

Now, we know that this side is the "equal" side of the isosceles triangle. Therefore, we can know that the total perimeter is:

Based on the description of this question, you can draw your triangle as such.  We can do this thanks to the nature of an isosceles triangle:

Now, you know that the area of a triangle is defined as:

So, for our data, we can say:

Solving for , we get:

Thus, .

Now, for our little triangle on the right, we can draw:

Using the Pythagorean Theorem, we know that the other side is:

This can be simplified to:

Now, we know that this side is the "equal" side of the isosceles triangle. Therefore, we can know that the total perimeter is:

Based on the description of your triangle, you can draw the following figure:

You can do this because you know:

1. The two equivalent sides are given.
2. Since a triangle is  degrees, you have only  or  degrees left for the two angles of equal size. Therefore, those two angles must be  degrees and  degrees. 

Now, based on the properties of an isosceles triangle, you can draw the following as well:

Based on your standard reference  triangle, you know:

Therefore,  is .

This means that  is , and the total base of the triangle is .

Now, the area of the triangle is:

 or  

Based on the description of your triangle, you can draw the following figure:

You can do this because you know:

1. The two equivalent sides are given.
2. Since a triangle is  degrees, you have only  or  degrees left for the two angles of equal size. Therefore, those two angles must be  degrees and  degrees.

Now, based on the properties of an isosceles triangle, you can draw the following as well:

Based on your standard reference  triangle, you know:

Therefore,  is .

This means that  is  and the total base of the triangle is .

Therefore, the perimeter of the triangle is:

Because the interior angles of a triangle add up to , and two of Triangle A's interior angles measure , we must simply add the two given angles and subtract from  to find the missing angle:

Therefore, the missing angle (and the smallest) from Triangle A measures . If the two triangles are similar, their interior angles must be congruent, meaning that the smallest angle is Triangle B is also .

The side measurements presented in the question are not needed to find the answer!

We must first find the length of the congruent sides in Triangle A. We do this by setting up a right triangle with the base and the height, and using the Pythagorean Theorem to solve for the missing side (). Because the height line cuts the base in half, however, we must use  for the length of the base's side in the equation instead of . This is illustrated in the figure below:

Using the base of  and the height of , we use the Pythagorean Theorem to solve for :

Therefore, the two congruent sides of Triangle A measure ; however, the question asks for the two congruent sides of Triangle B. In similar triangles, the ratio of the corresponding sides must be equal. We know that the base of Triangle A is  and the base of Triangle B is . We then set up a cross-multiplication using the ratio of the two bases and the ratio of  to the side we're trying to find (), as follows:

Therefore, the length of the congruent sides of Triangle B is .