Question

$\overline{bd}$ bisects $\angle abc$
reason:
given

$d$ is the midpoint of $\overline{ac}$
reason:
given

$\angle abd \cong \angle cbd$
reason:
select reason

$\overline{ad} \cong \overline{cd}$
reason:
select reason

$\angle adb \cong \angle cdb$
reason:
select reason

$\triangle abd \cong \triangle cbd$
reason:
select reason

Explanation:

Step1: Define angle bisector property

An angle bisector splits an angle into two congruent angles. Since $\overline{BD}$ bisects $\angle ABC$, this means $\angle ABD \cong \angle CBD$.

Step2: Define midpoint property

A midpoint divides a segment into two congruent segments. Since $D$ is the midpoint of $\overline{AC}$, this means $\overline{AD} \cong \overline{CD}$.

Step3: Define right angle congruence

The diagram shows $\angle ADB$ and $\angle CDB$ are right angles (marked with right angle symbols). All right angles are congruent, so $\angle ADB \cong \angle CDB$.

Step4: Prove triangle congruence

We have $\angle ABD \cong \angle CBD$, $\overline{BD} \cong \overline{BD}$ (reflexive property, common side), and $\angle ADB \cong \angle CDB$. This matches the Angle-Side-Angle (ASA) congruence criterion, so $\triangle ABD \cong \triangle CBD$.

Answer:

1. For $\boldsymbol{\angle ABD \cong \angle CBD}$: Reason is Definition of Angle Bisector
2. For $\boldsymbol{\overline{AD} \cong \overline{CD}}$: Reason is Definition of Midpoint
3. For $\boldsymbol{\angle ADB \cong \angle CDB}$: Reason is All Right Angles are Congruent
4. For $\boldsymbol{\triangle ABD \cong \triangle CBD}$: Reason is Angle-Side-Angle (ASA) Congruence Postulate