In the case of a triangle with one of the sides extended we have what is called an external supplementary angle (in this problem this angle is C degrees). It is "External" because it is not one of the three angles that makes up the triangle. "Supplementary" because it sits on a straight line and together with the adjacent interior angle it adds to 180.

Let's call the final angle inside the triangle D degrees. So inside the triangle we have angles of measures A degrees B degrees and D degrees. The external supplementary angle is C degrees. From here on I will just call them angles A B D and C.

With the supplementary angle rule we know that any two angles that make a straight line are equal to 180 degrees. So C + D = 180.

But it is also true that the angles that make up a triangle equal 180 degrees. So A + B + D = 180.

Since they each equal 180 we can set these equations equal to each other so that A + B + D = C + D. Subtract D from each side and you have A + B = C

The rule that takes you directly to "A + B = C" is as follows, "The measure of an external supplementary angle is equal to the sum of the two internal non-adjacent angles."

Clearly if A + B = C then quantity A equals zero.