Question

move reasons to the table to complete the proof that ∠bad≅∠bcd.

statement	reason
bd is an angle bisector of ∠abc	given
∠abd≅∠cbd	definition of angle bisector
bd = bd
△abd≅△cbd
∠bad≅∠bcd	corresponding parts of congruent triangles are congruent

symmetric property of congruence
sas congruence theorem
reflexive property of congruence
sss congruence theorem

Explanation:

Step1: Identify the reason for $BD = BD$

The side $BD$ is common to both $\triangle ABD$ and $\triangle CBD$. The property that a quantity is equal to itself is the reflexive property. So the reason for $BD = BD$ is the Reflexive Property of Congruence.

Step2: Identify the reason for $\triangle ABD\cong\triangle CBD$

We know that $AB = BC$ (given), $\angle ABD\cong\angle CBD$ (definition of angle - bisector) and $BD = BD$ (reflexive property of congruence). Two sides and the included angle of $\triangle ABD$ are congruent to two sides and the included angle of $\triangle CBD$. So, by the SAS (Side - Angle - Side) Congruence Theorem, $\triangle ABD\cong\triangle CBD$.

Answer:

Statement	Reason
$\overline{BD}$ is an angle bisector of $\angle ABC$	Given
$\angle ABD\cong\angle CBD$	Definition of Angle Bisector
$BD = BD$	Reflexive Property of Congruence
$\triangle ABD\cong\triangle CBD$	SAS Congruence Theorem
$\angle BAD\cong\angle BCD$	Corresponding Parts of Congruent Triangles are Congruent