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