Question

prove: △abd≅△cbd
statements

1. $overline{ab}congoverline{cb}$
2. $overline{bd}$ bisects ∠abc
3. ∠abd≅∠cbd
4. $overline{bd}congoverline{bd}$
5. △abd≅△cbd

reasons

1. given
2. given

3.
4.
5.

Explanation:

Step1: Definition of angle - bisector

If a ray bisects an angle, it divides the angle into two congruent angles. Since $\overline{BD}$ bisects $\angle ABC$ (given), then $\angle ABD\cong\angle CBD$.

Step2: Reflexive property of congruence

Any segment is congruent to itself. So, $\overline{BD}\cong\overline{BD}$.

Step3: SAS (Side - Angle - Side) congruence criterion

We have $\overline{AB}\cong\overline{CB}$ (given), $\angle ABD\cong\angle CBD$ (from step 1), and $\overline{BD}\cong\overline{BD}$ (from step 2). By the SAS congruence criterion, $\triangle ABD\cong\triangle CBD$.

Answer:

1. Reason: Definition of angle - bisector
2. Reason: Reflexive property of congruence
3. Reason: SAS congruence criterion