4. The condition that can determine quadrilateral $ABCD$ is a rhombus is ( ).
(A) Diagonal $AC$ bisects diagonal $BD$, and $AC$ $\perp BD$
(B) Diagonal $AC$ bisects diagonal $BD$, and $\angle A$ $=\angle C$
(C) Diagonal $AC$ bisects diagonal $BD$, and bisects $\angle A, \angle C$
(D) Diagonal $AC$ bisects $\angle A, \angle C$, and $\angle A$ $=\angle C$
4. (D) .

As shown in the figure (referred to as a pen shape),
$AC$ bisects $BD$, $AC \perp BD$, $AC$ also bisects $\angle A, \angle C$, so we can rule out (A) and (C); a parallelogram that is not a rhombus can rule out (B); according to the method of determining a rhombus, we can prove that (D) is correct.