The measure of an exterior angle of a triangle is the sum of the measures of its remote interior angles, so 

In an isosceles triangle, the base angles are the same. Also, the three angles of a triangle add up to .

So, subtract the vertex angle from . You get .

Because there are two base angles you divide  by , and you get .

The measure of an exterior angle of a triangle is the sum of the measures of its remote interior angles, so 

.

We also have the following constraints:

Then, by the addition property of inequalities,

Therefore, the measure of  must fall in that range. Of the given choices, only  falls in that range.

The measure of an exterior angle of a triangle is the sum of the measures of its remote interior angles, so 

or 

Therefore, the maximum value of  is the least possible value of  subtracted from the greatest possible value of :

The minimum value of  is the greatest possible value of  subtracted from the least possible value of :

Therefore, 

Since all of the choices fall in this range, all are possible measures of .

All the angles in a triangle must add up to .

All the angles in a triangle must add up to 

All the angles in a triangle must add up to .

By the Isosceles Triangle Theorem, an isosceles triangle has two congruent interior angles. There are two possible scenarios if one angle has measure :

Scenario 1: The other two angles are congruent to each other. The degree measures of the interior angles of a triangle total , so if we let  be the common measure of those angles:

This makes (b) a possible answer.

Scenario 2: One of the other angles measures  also, making (c) a possible answer. The degree measure of the third angle is

,

making (a) a possible answer. Therefore, the correct choice is (a), (b), or (c).

A scalene triangle has three sides of different measure, so, by way of the Converse of the Isosceles Triangle Theorem, each angle is of different measure as well. We can therefore eliminate  immediately. 

Also, if the triangle also has a  angle, then, since the total of the degree measures of the angles is , it follows that the third angle has measure

.

Therefore, the triangle has two angles that measure the same, and  can be eliminated.

Similarly, if the triangle also has a  angle, then, since the total of the degree measures of the angles is , it follows that the third angle has measure

.

The triangle has two angles that measure . This choice can be eliminated.

 can be eliminated, since the third angle would have measure

,

an impossible situation since angle measures must be positive.

The remaining possibility is . This would mean that the third angle has measure

.

The three angles have different measures, so the triangle is scalene.  is the correct choice.

Above is the figure described.

The measures of the interior angles of a triangle total , so the measure of  is

Since  bisects this angle, 

and