Q:
What are the angles of the triangle ABC, if the sum of the angles A and B is equal to 100 degrees, the sum of the angles B and C equal to 120 degrees?

The triangle and the angles 
We have two sum of two angles:
A + B = 100
B + C = 120
The sum of angles in a triangle is 180 degrees.
A + B + C = 180
Substitute in this formula the sum of two angles and find the third angle:
100 + C = 180
C = 180  100
C = 80
The second sum of the angles substitute angle C:
B + 80 = 120
B = 120  80
B = 40
The first sum of angles substitute angle B:
A + 40 = 100
A = 100  40
A = 60
A: The angles of the triangle are equal: A=60 degrees, B=40 degrees, C=80 degrees.
If we substitute in the second sum angles of a triangle, then the solution would be:
A + 120 = 180
A = 180  120
A = 60
60 + B = 100
B = 100  60
B = 40
60 + 40 + C = 180
C = 180  60  40
C = 80
The third variant of the decision:
We find the angle A of the first sum.
A + B = 100
A = 100  B
We find the angle C of the second sum.
B + C = 120
C = 120  B
Substitute the found angles to the general formula:
A + B + C = 180
(100  B) + B + (120  B) = 180
100  B + B + 120  B = 180
B  2B = 180  100  120
B = 40
B = 40
A = 100  40
A = 60
C = 120  40
C = 80
Conclusion: if the problem is solved correctly, the result does not depend on the method of solution.