Question

given $overline{bd}$ bisects $overline{ac}$, complete the flowchart proof below.
$overline{bd}$ bisects $overline{ac}$
reason: given

Explanation:

Step1: Define bisecting property

Since $\overline{BD}$ bisects $\overline{AC}$, we have $AD = DC$. Reason: Definition of a bisector.

Step2: Identify congruent angles

We may assume $\angle ADB=\angle CDB$ (if $BD$ is also an angle - bisector or by vertical - angle or other given conditions. If not given, we can't assume this yet). But if we assume $\angle ADB$ and $\angle CDB$ are right - angles (or if it's given), or if we have other angle - congruence conditions. Let's assume for the sake of congruence proof that we have $\angle ADB=\angle CDB$.

Step3: Identify common side

$BD = BD$ by the reflexive property of equality.

Step4: Prove triangle congruence

If we have the side - angle - side (SAS) congruence criterion (i.e., $AD = DC$, $\angle ADB=\angle CDB$, $BD = BD$), then $\triangle ADB\cong\triangle CDB$.

Answer:

We can prove $\triangle ADB\cong\triangle CDB$ using the SAS congruence criterion if we have the appropriate angle - congruence condition along with the given side - bisecting property and the reflexive property of the common side $BD$. Without full information about angles, we can only state the steps based on the given side - bisecting fact and the general congruence - proof approach. If we assume $\angle ADB=\angle CDB$ (by given right - angle or other angle - congruence condition), then $\triangle ADB\cong\triangle CDB$ by SAS ($AD = DC$ from $BD$ bisecting $AC$, $\angle ADB=\angle CDB$, and $BD = BD$).