Question

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Explanation:

Step1: Identify given congruent sides

We know $\overline{AB} \cong \overline{CB}$ (given).

Step2: Recall angle bisector result

$\overline{BD}$ bisects $\angle ABC$, so $\angle ABD \cong \angle CBD$, and $\overline{BD} \cong \overline{BD}$ (reflexive property).

Step3: Apply SAS congruence

By SAS, the triangles formed by the bisector are congruent. The triangles are $\triangle ABD$ and $\triangle CBD$.

Step4: Verify match to options

This congruence matches the last option, and it is the statement needed for the SAS postulate step (step 4 in the proof) to lead to CPCTC.

Answer:

$\boldsymbol{\triangle ABD \cong \triangle CBD}$