  For a given inverse semigroup, one can associate an \'etale groupoid which is
called the universal groupoid. Our motivation is studying the relation between
inverse semigroups and associated \'etale groupoids. In this paper, we focus on
congruences of inverse semigroups, which is a fundamental concept to consider
quotients of inverse semigroup. We prove that a congruence of an inverse
semigroup induces a closed invariant set and a normal subgroupoid of the
universal groupoid. Then we show that the universal groupoid associated to a
quotient inverse semigroup is described by the restriction and quotient of the
original universal groupoid. Finally we compute invariant sets and normal
subgroupoids induced by special congruences including abelianization.
