Isosceles triangle is a regular polygon if its base equals in length to its sides.
An isosceles triangle is, by definition, the one with two sides equal in length to each other (congruent). The third, called the base, can have any length possible.
Regular polygon, by definition, is the one with all sides and all interior angles equal in measure (congruent).
Therefore, for an isosceles triangle to be regular polygon, it needs the base to be equal in length to each of the other two sides that already are congruent to each other.
Congruence of all interior angles in case when all three sides are congruent is a standard theorem that is proven in the course of elementary geometry.