The isosceles triangle has two congruent sides and two congruent angles.  

The interior angles will sum to 180 degrees.

Let the unknown angles be .  These angles are congruent to each other because the triangle is isosceles.

Divide by 2 on both sides.

The answer is:  

 is indicated to be isosceles, with , so, by the isosceles triangle theorem, 

.

Since the measures of the interior angles of a triangle total , 

Substitute  for  and  for , then solve for :

Subtract  from both sides:

Multiply both sides by :

 is an exterior angle of ; its measure is the sum of those of its remote interior angle,  and , so 

Substitute and add:

.

An isosceles has two equivalent base angles.  We can set up an equation such that both angles with the vertex angle add up to 180 degrees.

Solve for , which is a base angle.  Subtract 30 from both sides.

Divide by 2 on both sides.

The answer is:  

Start by figuring out the degree measurements of each angle.

Since we know that the angles of a triangle must add up to , we can write the following equation:

Solve for .

Now, we know the measure of the angles:

Since each angle of the triangle is different, this means the the legs of the triangle must also be different. Thus, this is a scalene triangle.

By the Isosceles Triangle Theorem, an isosceles triangle - a triangle with at least two sides of equal length - must have at least two angles of equal degree measure. The choice  can therefore be immediately eliminated.

Also, the degree measures must total , so add the measures in each group to find the set that conforms to this condition:

:

:

:

The last group is the correct choice.

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.