Question

bd bisects ∠abc
reason: given
∠abd ≅ ∠cbd
reason: select reason.
bd ≅ bd
reason: select reason.
∠a ≅ ∠c
reason: select reason.
△abd ≅ △cbd
reason: select reason.
search for reason...
given
reflexive property
vertical angles are congruent

Explanation:

Step1: Reason for ∠ABD ≅ ∠CBD

Since BD bisects ∠ABC, by the definition of an angle bisector, it divides ∠ABC into two equal angles. So the reason is "Definition of angle bisector".

Step2: Reason for ∠A ≅ ∠C

This is likely given (assuming from the problem - solving context in geometry proofs where some angles can be given as congruent). So the reason is "Given".

Step3: Reason for BD ≅ BD

By the reflexive property (a segment is congruent to itself). So the reason is "Reflexive Property".

Step4: Reason for △ABD ≅ △CBD

We have ∠ABD ≅ ∠CBD (from Step 1), ∠A ≅ ∠C (from Step 2), and BD ≅ BD (from Step 3). By the AAS (Angle - Angle - Side) congruence criterion. So the reason is "AAS".

Answer:

• For ∠ABD ≅ ∠CBD: Definition of angle bisector
• For ∠A ≅ ∠C: Given
• For BD ≅ BD: Reflexive Property
• For △ABD ≅ △CBD: AAS