Given that $CA$ bisects $\angle BAD$ and $CA$ bisects $\angle BCD$, prove that $\triangle ABC \cong \triangle ADC$.

1. Problem Description

Given that $CA$ bisects $\angle BAD$ and $CA$ bisects $\angle BCD$, prove that $\triangle ABC \cong \triangle ADC$.

2. Solution Steps

Step 1: Given

$CA$ bisects $\angle BAD$

$CA$ bisects $\angle BCD$

Reason: Given

Step 2:

$\angle BAC \cong \angle DAC$

Reason: Definition of angle bisector

Step 3:

$\angle BCA \cong \angle DCA$

Reason: Definition of angle bisector

Step 4:

$AC \cong AC$

Reason: Reflexive Property of Congruence

Step 5:

$\triangle ABC \cong \triangle ADC$

Reason: Angle-Side-Angle (ASA) Congruence Postulate

3. Final Answer

$\triangle ABC \cong \triangle ADC$