  In this article, we introduce the notion of pre-$(n+2)$-angulated categories
as higher dimensional analogues of pre-triangulated categories defined by
Beligiannis-Reiten. We first show that the idempotent completion of a
pre-$(n+2)$-angulated category admits a unique structure of
pre-$(n+2)$-angulated category. Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be
an $n$-exangulated category and $\mathscr{X}$ be a strongly functorially finite
subcategory of $\mathscr{C}$. We then show that the quotient category
$\mathscr{C}/\mathscr{X}$ is a pre-$(n+2)$-angulated category.These results
allow to construct several examples of pre-$(n+2)$-angulated categories.
Moreover, we also give a necessary and sufficient condition for the quotient
$\mathscr{C}/\mathscr{X}$ to be an $(n+2)$-angulated category.
