Question

$\overline{ac}$ bisects $\angle bad$
reason:
given

$\overline{ac}$ bisects $\angle bcd$
reason:
given

$\angle bac \cong \angle dac$
reason:
an angle bisector divides an angle into two congruent angles

$\angle bca \cong \angle dca$
reason:
select reason

$\overline{bc} \cong \overline{dc}$
reason:
select reason

$\triangle abc \cong \triangle adc$
reason:
sss

Explanation:

Step1: Reason for ∠BCA ≅ ∠DCA

Since $\overline{AC}$ bisects $\angle BCD$ (given), an angle bisector splits an angle into two congruent angles. So the reason is "An angle bisector divides an angle into two congruent angles".

Step2: Reason for $\overline{BC} \cong \overline{DC}$

From $\triangle ABC \cong \triangle ADC$ (SSS congruence), corresponding parts of congruent triangles are congruent (CPCTC). So $\overline{BC}$ and $\overline{DC}$ as corresponding sides are congruent.

Answer:

Reason for $\angle BCA \cong \angle DCA$: An angle bisector divides an angle into two congruent angles
Reason for $\overline{BC} \cong \overline{DC}$: Corresponding parts of congruent triangles are congruent (CPCTC)