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Although the two words have almost the same meaning, they are different. Let's start by defining the two terms.
A complement is a word used when you want to describe the compatibility of their things. Mind you, the word complement should not be mistaken with the word compliment which means to say something nice to someone
An example of how to use the word complement
That red tie complements your suit
The spicy sauce complemented the main dish
A supplement is refferd to something that is added to give extra support or boost
For example
I'm taking a part time job to supplement my main income
Note that although both of them sounds the same they are actually different. Complement deals with accompaniment and supplement deals with addition.
