Let
k
\mathsf {k}
be a field, let
S
S
be a bigraded
k
\mathsf {k}
-algebra, and let
S
Δ
S_\Delta
denote the diagonal subalgebra of
S
S
corresponding to
Δ
=
{
(
c
s
,
e
s
)
|
s
∈
Z
}
\Delta = \{ (cs,es) \; | \; s \in \mathbb {Z} \}
. It is known that the
S
Δ
S_\Delta
is Koszul for
c
,
e
≫
0
c,e \gg 0
. In this article, we find bounds for
c
,
e
c,e
for
S
Δ
S_\Delta
to be Koszul when
S
S
is a geometric residual intersection. Furthermore, we also study the Cohen-Macaulay property of these algebras. Finally, as an application, we look at classes of linearly presented perfect ideals of height two in a polynomial ring, show that all their powers have a linear resolution, and study the Koszul and Cohen-Macaulay properties of the diagonal subalgebras of their Rees algebras.