Suppose that we are trying to define a mathematical object $M$. The statement of the definition generally takes the form (or some of its equivalent variant),
A mathematical object is said to be $M$ $\color{red}{\text{if}}$ it satisfies some property or properties, say $P$.
It is said that the inclusion of "$\color{red}{\text{if}}$" in the definitions can be easily replaced by "$\color{blue}{\text{iff}}$". In other words, the above definition is equivalent to saying,
A mathematical object is said to be $M$ $\color{blue}{\text{iff}}$ it satisfies some property or properties, say $P$.
In Fraleigh's book on Abstract Algebra it says,
But I think that a definition is only an $\color{red}{\text{if}}$-type statement. Take for example the definition of a (Euclidean) triangle
A geometric object is said to be a triangle $\color{blue}{\text{iff}}$ it has three sides.
This definition includes two parts,
A geometric object is said to be a triangle if it has three sides.
and,
If a geometric object is a triangle then has three sides.
But the second part is clearly nonsense if we assume that we have exactly one definition of a triangle for in the second part we have to know what a triangle actually is and by our assumption the definition of a triangle is provided by first part.
So, my question is,
Why is every definition an "if and only if" type statement ?