Prove: $\angle A B D$ and $\angle E B C$ are complementary
\begin{tabular}{|l|l|}
\hline \multicolumn{1}{|c|}{ Statements } & \\
\hline 1. $\overrightarrow{B D} \perp \overrightarrow{B C}$ & 1. \\
\hline 2. $\angle D B C$ is a right angle & 2. \\
\hline 3. $m \angle D B C=90^{\circ}$ & 3. \\
\hline 4. $m \angle D B E+m \angle E B C=m \angle D B C$ & 4. \\
\hline 5. $m \angle D B E+m \angle E B C=90^{\circ}$ & 5. \\
\hline 6. $\angle A B D \cong \angle D B E$ & 6. \\
\hline 7. $m \angle A B D=m \angle D B E$ & 7. \\
\hline 8. $m \angle A B D+m \angle E B C=90^{\circ}$ & 8. \\
\hline 9. $\angle A B D$ and $\angle E B C$ are complementary & 9. \\
\hline
\end{tabular}

Step 1 ：Given that \(\overrightarrow{B D} \perp \overrightarrow{B C}\), it implies that \(\angle D B C\) is a right angle.

Step 2 ：Since \(\angle D B C\) is a right angle, its measure is \(90^\circ\).

Step 3 ：By the Angle Addition Postulate, \(m \angle D B E+m \angle E B C=m \angle D B C\).

Step 4 ：Substituting the known value, we get \(m \angle D B E+m \angle E B C=90^\circ\).

Step 5 ：Given that \(\angle A B D \cong \angle D B E\), their measures are equal, i.e., \(m \angle A B D=m \angle D B E\).

Step 6 ：Substituting the equal measures in the equation, we get \(m \angle A B D+m \angle E B C=90^\circ\).

Step 7 ：This implies that \(\angle A B D\) and \(\angle E B C\) are complementary.

Step 8 ：\(\boxed{\text{Therefore, the statement that } \angle A B D \text{ and } \angle E B C \text{ are complementary is true.}}\)