The given information is actually inconclusive.

By the Angle-Angle Similarity Postulate, if two pairs of corresponding angles of a triangle are congruent, the triangles themselves are similar. Therefore, we seek to prove two of the following three angle congruence statements:

 and  are a pair of vertical angles, having the same vertex and having sides opposite each other. As such, . 

, but this is not one of the statements we need to prove. Also, without further information - for example, whether  and  are parallel, which is not given to us - we have no way to prove either of the other two necessary statements. 

The correct response is "false".

As we are establishing whether or not , then , , and  correspond respectively to , , and .

 is an isosceles triangle, so it must have two congruent angles.  has measure , so either  has this measure,  has this measure, or . If we examine the second case, it immediately follows that . One condition of the similarity of triangles is that all pairs of corresponding angles be congruent; since there is at least one case that violates this condition, it does not necessarily follow that  . This makes the correct response "false".

As we are establishing whether or not , then , , and  correspond respectively to , , and .

A triangle is equilateral (having three sides of the same length) if and only if it is also equiangular (having three angles of the same measure, each of which is  ). It follows that all angles of both triangles measure .

Specifically,  and , making two pairs of corresponding angles congruent. By the Angle-Angle Similarity Postulate, it follows that , making the correct answer "true".

The sum of the measures of the interior angles of a triangle is , so

Also,

,

so

By similar reasoning, it holds that

Since , by substitution,

Therefore, 

,

or 

This, along with the statement that , sets up the conditions of the Angle-Angle Similarity Postulate - if two angles of one triangle are congruent to the two corresponding angles of another triangle, the two triangles are similar. It follows that 

.

However, congruence cannot be proved, since at least one side congruence is needed to prove this. This is not given in the problem. 

Therefore, statement (a) must hold, but not necessarily statement (b).

A triangle has degrees.  An isoceles triangle has one vertex angle and two congruent base angles.

Let = the base angle and  = vertex angle

So the equation to solve becomes 

or

so the base angle is and the vertex angle is and the difference is .

Isosceles triangles always have two equivalent interior angles, and all three interior angles of any triangle always have a sum of  degrees. 

Thus, the solution is:

Isosceles triangles always have two equivalent interior angles, and all three interior angles of any triangle always have a sum of  degrees. Since this is an obtuse isosceles triangle, the two missing angles must be acute angles.  

Thus, the solution is:

Isosceles triangles always have two equivalent interior angles, and all three interior angles of any triangle always have a sum of  degrees.

Thus, the solution is:

Isosceles triangles always have two equivalent interior angles, and all three interior angles of any triangle always have a sum of  degrees. Since this is an obtuse isosceles triangle, the two missing angles must be acute angles. 

The solution is:



However,  degrees is the measurement of both of the acute angles combined.

Each individual angle is .

Isosceles triangles always have two equivalent interior angles, and all three interior angles of any triangle always have a sum of  degrees. Since this is an acute isosceles triangle, all of the interior angles must be acute angles.

The solution is: