The problem states that line segment $CA$ bisects angle $BAD$ and angle $BCD$. We need to prove that triangle $ABC$ is congruent to triangle $ADC$.

1. Problem Description

The problem states that line segment $CA$ bisects angle $BAD$ and angle $BCD$. We need to prove that triangle $ABC$ is congruent to triangle $ADC$.

2. Solution Steps

Step 1: $CA$ bisects $\angle BAD$ and $CA$ bisects $\angle BCD$. Reason: Given.

Step 2: $\angle BAC \cong \angle DAC$ and $\angle BCA \cong \angle DCA$. Reason: Definition of angle bisector. An angle bisector divides an angle into two congruent angles.

Step 3: $AC \cong AC$. Reason: Reflexive Property of Congruence.

Step 4: $\triangle ABC \cong \triangle ADC$. Reason: Angle-Side-Angle (ASA) Congruence Postulate. We have $\angle BAC \cong \angle DAC$, $AC \cong AC$, and $\angle BCA \cong \angle DCA$.

3. Final Answer

$\triangle ABC \cong \triangle ADC$