Question

1. write a two - column proof.

given: $overrightarrow{bd}$ bisects $angle cbe$.
prove: $angle abdcongangle fbd$

Explanation:

Step1: Recall angle - bisector definition

Since $\overrightarrow{BD}$ bisects $\angle CBE$, we have $\angle CBD=\angle DBE$.

Step2: Use linear - pair and angle - addition properties

$\angle ABD + \angle CBD=180^{\circ}$ and $\angle FBD+\angle DBE = 180^{\circ}$ (linear pairs). So, $\angle ABD+\angle CBD=\angle FBD+\angle DBE$.

Step3: Substitute equal angles

Because $\angle CBD=\angle DBE$, we can substitute $\angle CBD$ for $\angle DBE$ in the equation $\angle ABD+\angle CBD=\angle FBD+\angle DBE$. Then we get $\angle ABD=\angle FBD$, which means $\angle ABD\cong\angle FBD$.

Answer:

The proof is completed as above to show $\angle ABD\cong\angle FBD$.