Question

ac bisects ∠bcd reason: given ∠bca≅∠dca reason: an angle bisector divides an angle into two congruent angles ∠bac≅∠dac reason: given bc≅dc reason: reflexive property △abc≅△adc reason: sas

Explanation:

Step1: Identify given information

We are given that $\overline{AC}$ bisects $\angle BCD$, so $\angle BCA\cong\angle DCA$ (by the definition of an angle - bisector). Also given is $\angle BAC\cong\angle DAC$, and $\overline{BC}\cong\overline{DC}$ by the reflexive property.

Step2: Apply congruence criterion

In $\triangle ABC$ and $\triangle ADC$, we have two pairs of congruent angles ($\angle BCA\cong\angle DCA$ and $\angle BAC\cong\angle DAC$) and a pair of congruent sides ($\overline{BC}\cong\overline{DC}$). The Side - Angle - Side (SAS) congruence criterion states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. Here, the included angles for the congruent sides are the given congruent angles. So, $\triangle ABC\cong\triangle ADC$ by SAS.

Answer:

The reasoning for $\triangle ABC\cong\triangle ADC$ is correct as it is based on the given information and the application of the SAS congruence criterion.