Question

ac ⊥ bd
reason:
given

∠adb and ∠cdb are right angles
reason:
perpendicular lines form right angles

bd bisects ac
reason:
given

∠adb ≅ ∠cdb
reason:
select reason

ad ≅ cd
reason:
select reason

∠a ≅ ∠c
reason:
select reason

△abd ≅ △cbd
reason:
select reason

Explanation:

Step1: Justify right angles congruence

All right angles are congruent. So $\angle ADB \cong \angle CDB$ because they are both right angles.

Step2: Justify segment congruence from bisector

A bisector divides a segment into two congruent parts. Since $\overline{BD}$ bisects $\overline{AC}$, $\overline{AD} \cong \overline{CD}$.

Step3: Justify given angle congruence

$\angle A \cong \angle C$ is marked as given in the diagram (the tick marks on the angles indicate this).

Step4: Justify triangle congruence

We have $\angle A \cong \angle C$, $\overline{AD} \cong \overline{CD}$, and $\angle ADB \cong \angle CDB$. This matches the ASA (Angle-Side-Angle) congruence criterion.

Answer:

1. For $\angle ADB \cong \angle CDB$: Reason = All right angles are congruent
2. For $\overline{AD} \cong \overline{CD}$: Reason = Definition of a segment bisector
3. For $\angle A \cong \angle C$: Reason = Given
4. For $\triangle ABD \cong \triangle CBD$: Reason = ASA (Angle-Side-Angle) Triangle Congruence Postulate