For this problem, remember that the sum of the degrees in a triangle is .

This means that .

Plug in our given values to solve:

The sum of the two smallest sides is less than the greatest side:

By the Triangle Inequality, this triangle cannot exist.

The key to solving this problem lies in the geometric fact that a triangle possesses a total of  between its interior angles.  Therefore, one can calculate the measure of  and then find the measure of its supplementary angle, .

 and  are supplementary, meaning they form a line with a measure of .

One could also solve this problem with the knowledge that the sum of the exterior angle of a triangle is equal to the sum of the two interior angles opposite of it.

The key to solving this problem lies in the geometric fact that a triangle possesses a total of  between its interior angles.  Therefore, one can calculate the measure of  and then find the measure of its supplementary angle, .

 and  are supplementary, meaning they form a line with a measure of .

One could also solve this problem with the knowledge that the sum of the exterior angle of a triangle is equal to the sum of the two interior angles opposite of it.

The sum of the angles of any triangle is always  degrees. Since the third angle will make up the difference between  and the sum of the other two angles, add the other two angles together and subtract this sum from .

Sum of the two given angles:  degrees

Difference between  and this sum:  degrees

  If angle A is 41°, then angle C must also be 41°, since AB=BC.  So, the sum of these 2 angles is:

41° + 41° = 82°

Since the sum of the angles in a triangle is 180°, you can find out the measure of the remaining angle by subtracting 82 from 180:

180° - 82° = 98°

Recall that the sum of the angles of a triangle is  degrees. Since we are given two angles, we can then find the third. Call our missing angle . 

We combine the like terms on the left. 

Subtract  from both sides.

Thus, we have that our missing angle is  degrees. 

We know that the sum of the angles of a triangle must add up to  degrees. The two given angles sum to  degrees. Thus, a triangle cannot be formed.

To solve this question, you need to remember that the sum of the angles in a triangle is 180°. You also need to remember supplementary angles. If you know what ∠ DCE is, you also know what ∠ ECA is. Hence you know two angles of the triangle, 180°-80°-50°= 50°. 

Since the triangle is isosceles, we know that 2 of the angles (that sum up to 180) must be equal. The question states that the noncongruent angles average 55°, thus providing us with a system of two equations:

Solving for x and y by substitution, we get x = 70° and y = 40° (which average out to 55°).

70 + 70 + 40 equals 180 also checks out.

Since 70° is not an answer choice for us, we know that the 40° must be one of the angles.