Question

ac bisects ∠bcd
reason: given
∠bca≅∠dca
reason: select reason
∠bac≅∠dac
reason: select reason
bc≅dc
reason: select reason
△abc≅△adc
reason: select reason

Explanation:

Step1: Definition of an angle - bisector

If a ray bisects an angle, it divides the angle into two congruent angles. Since $\overline{AC}$ bisects $\angle BCD$, by the definition of an angle - bisector, $\angle BCA\cong\angle DCA$.

Step2: No information for $\angle BAC\cong\angle DAC$

There is no given information to support $\angle BAC\cong\angle DAC$. We cannot assume this without further given conditions.

Step3: No information for $\overline{BC}\cong\overline{DC}$

There is no given information to support $\overline{BC}\cong\overline{DC}$. We cannot assume this without further given conditions.

Step4: For $\triangle ABC\cong\triangle ADC$

If we only have $\angle BCA\cong\angle DCA$ and the common side $\overline{AC}=\overline{AC}$, we do not have enough information to prove $\triangle ABC\cong\triangle ADC$. To prove the triangles congruent, we would need more information such as another pair of sides or angles.

Answer:

For $\angle BCA\cong\angle DCA$, the reason is "Definition of angle - bisector". For $\angle BAC\cong\angle DAC$, there is no valid reason with the given information. For $\overline{BC}\cong\overline{DC}$, there is no valid reason with the given information. For $\triangle ABC\cong\triangle ADC$, there is not enough information to prove congruence with the given data.