For all triangles, the sum of the three angles is equal to  degrees. Therefore, if you are given two angles and you need to solve for the third one, you need to add the two angles you know and subtract that from . Because  and because , the third angle has a measure of  degrees. 

First we need to recall that whenever we add up all 3 angles of any given triangle, the sum will always be .

In an isosceles triangle two of the angles are congruent. Since we are told that one of the angles of our triangle is  we know that this is an obtuse triangle, since 120 is greater than 90.

We need to subtract 120 from 180 to find the remainder of the triangle which is 

Since we are working with an isosceles triangle, we know that the remaining two angles are going to be congruent. To find the degree of the angles we simply divide 60 by 2. Our answer is; both angles are 

 is simply a restatement of , since the names of the corresponding vertices of the similar triangles are still in the same relative positions.

 is a consequence of , since corresponding angles of similar triangles are, by definition, congruent.

 is a consequence of , since corresponding sides of similar triangles are, by definition, in proportion.

However, similar triangles need not have congruent corresponding sides. Therefore, it does not necessarily follow that . This is the correct choice.

The similarity of two triangles implies nothing about the relationship of two angles of the same triangle. Therefore,  can be eliminated.

The similarity of two triangles implies that corresponding angles between the triangles are congruent. However, because of the positions of the letters,  in  corresponds to , not , in , so . The statement  can be eliminated.

Similarity of two triangles does not imply any congruence between sides of the triangles, so  can be eliminated.

Similarity of triangles implies that corresponding sides are in proportion.  and  in  correspond, respectively, to  and  in . Therefore, it follows that , and this statement is the correct choice.

Suppose . 

Corresponding angles of similar triangles are congruent, so . Also, , so, since  is a right angle, so is .

Corresponding sides of similar triangles are in proportion. Since 

, 

the similarity ratio of  to  is 3.

By the Pythagorean Theorem, since  is the hypotenuse of a right triangle with legs 6 and 8, its measure is 

.

 , so  is a true statement.

But , so  is false if the triangles are similar. This is the correct choice.

By the Pythagorean Theorem, since  is the hypotenuse of a right triangle with legs 6 and 8, its measure is 

.

The similarity ratio of  to  is

.

Likewise, 

Corresponding angles of similar triangles are congruent, so, since  is right, so is . This makes  and  the legs of a right triangle, so its area is half their product.

By the Pythagorean Theorem, since  is the hypotenuse of a right triangle with legs 6 and 8, its measure is 

.

The similarity ratio of  to  is

.

This can be used to find  and :

The area of  is therefore 

.

Since the two triangles are similar, we know that their corresponding sides must be in the same ratio to each other. Thus, we can write the following equation:

Now, solve for .

By the Pythagorean Theorem, if  are the legs of a right triangle and  is its hypotenuse, 

Substitute  and solve for :

The perimeter of the triangle is 

The hypotenuse of the right triangle can be calculated using the Pythagorean theorem:

Add the three sides: