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 

If  is isosceles, then by the Isosceles Triangle Theorem, two of its angles must be congruent. 

Case 1: 

Since  bisects  into two congruent angles, one of which must be , 

However, this is impossible, since  and  are two angles of the original triangle; their total measure is

Case 2: 

Then, since the degree measures of the interior angles of a triangle total ,

Since  bisects  into two congruent angles, one of which must be , 

and

Case 3: 

Then

, which is not possible.

Therefore, the only possible measure of  is .

The degree measures of the interior angles of a triangle total 180 degrees, so 

One angle measures 

The other two angles measure 

and 

.

We want the greatest of the three, or .

The figure referenced is below:

Since  is an isosceles right triangle, its acute angles - in particular,  - measure  each. Since this angle forms a linear pair with :

.

  is also isosceles, so, by the Isosceles Triangle Theorem, it has two congruent angles. Since  is obtuse, and no triangle has two obtuse angles:

.

Also,  is an exterior angle of , whose measure is equal to the sum of those of its two remote interior angles, which are the congruent angles . Therefore,

The perimeter is the length of all the sides added up.

Using the information given in the question,

Now, solve for side 3.

To find the perimeter of a triangle, add up all of its sides. 

Let  be the length of the third side.

The length of the third side is .