To explain this clearly, refer to the figure attached.
Angle addition, from the name itself, is the sum of the smaller angles to make up the bigger angle. In that case, the yellow angle plus the green angle, equals the maroon angle. As long as they have the same vertex (vertex O), you can add up angles that are side by side to obtain the value of the total angle. That is what the phrase "the whole is equal to the sum of its parts" means.
