Question

bd bisects ∠abc
reason: given
∠abd≅∠cbd
reason: select reason
∠a≅∠c
reason: select reason
bd≅bd
reason: select reason
△abd≅△cbd
reason: select reason

Explanation:

Step1: Definition of angle - bisector

If a ray bisects an angle, it divides the angle into two congruent angles. Since \(BD\) bisects \(\angle ABC\), we have \(\angle ABD\cong\angle CBD\).

Step2: Given information

The reason for \(\angle A\cong\angle C\) is given (assuming it is given in the problem - context).

Step3: Reflexive property of congruence

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

Step4: Angle - Angle - Side (AAS) congruence criterion

Since \(\angle A\cong\angle C\), \(\angle ABD\cong\angle CBD\), and \(\overline{BD}\cong\overline{BD}\), by the AAS congruence criterion, \(\triangle ABD\cong\triangle CBD\).

Answer:

• \(\angle ABD\cong\angle CBD\) Reason: Definition of angle - bisector
• \(\angle A\cong\angle C\) Reason: Given
• \(\overline{BD}\cong\overline{BD}\) Reason: Reflexive property of congruence
• \(\triangle ABD\cong\triangle CBD\) Reason: AAS (Angle - Angle - Side) congruence criterion