We are given that $\overline{CA}$ bisects $\angle BAD$ and $\overline{CA}$ bisects $\angle BCD$. We need to prove that $\triangle ABC \cong \triangle ADC$.

1. Problem Description

We are given that $\overline{CA}$ bisects $\angle BAD$ and $\overline{CA}$ bisects $\angle BCD$. We need to prove that $\triangle ABC \cong \triangle ADC$.

2. Solution Steps

Step 1: $\overline{CA}$ bisects $\angle BAD$ and $\overline{CA}$ bisects $\angle BCD$. (Given)

Step 2: $\angle BAC \cong \angle DAC$ and $\angle BCA \cong \angle DCA$. (Definition of angle bisector)

Step 3: $\overline{AC} \cong \overline{AC}$. (Reflexive Property)

Step 4: $\triangle ABC \cong \triangle ADC$. (ASA Congruence)

3. Final Answer

$\triangle ABC \cong \triangle ADC$