4. In quadrilateral $A B C D$, diagonal $A C$ is equal to the adjacent sides $A B$ and $A D$. If $\angle D A C$ is $k$ times $\angle C A B$ ($k$ is a real number), then $\angle D B C$ is ( ) of $\angle B D C$.
(A) $k$ times
(B) $2 k$ times
(C) $3 k$ times
(D) None of the above answers is correct
4. (A).

Given, as shown in the figure, points $B$, $C$, and $D$ are on a circle with center $A$ and radius $AB$.
$$
\begin{array}{l}
\because \angle D A C=k \angle C A B, \\
\text { and } \angle D B C=\frac{1}{2} \angle D A C, \\
\angle B D C=\frac{1}{2} \angle B A C, \\
\therefore \angle D B C=k \angle B D C,
\end{array}
$$