The prove, segment BD is congruent to segment DC. is achieved using SAS congruence theorem and CPCTC
How to complete the proof
The two column proof is written as follows
Statement Reason
AD is an angle bisector of BAC given
∠ 1 ≅ ∠ 2 definition of angle bisector
AB ≅ AC given
AD ≅ DA Reflexive property
Δ BAD ≅ Δ CAD SAS congruence theorem
BD ≅ DC CPCTC