Let k be a field, let S be a bigraded k-algebra, and let SΔ denote the diagonal subalgebra of S corresponding to Δ = {(cs, es) | s ∈ Z}. It is known that the SΔ is Koszul for c, e ≫ 0. In this article, we find bounds for c, e for SΔ to be Koszul when 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. © 2019 American Mathematical Society.