1. **State the problem:** Given that CA bisects \(\angle BAD\), \(AB \perp BC\), and \(AD \perp DC\), prove that \(\triangle ABC \cong \triangle ADC\). 2. **Identify given information and what to prove:** - \(CA\) bisects \(\angle BAD\) means \(\angle BAC = \angle DAC\). - \(AB \perp BC\) means \(\angle ABC = 90^\circ\). - \(AD \perp DC\) means \(\angle ADC = 90^\circ\). - We want to prove \(\triangle ABC \cong \triangle ADC\). 3. **Use the Angle-Side-Angle (ASA) congruence criterion:** - We have \(\angle BAC = \angle DAC\) (angle bisector). - \(AC\) is common side to both triangles. - \(\angle ABC = \angle ADC = 90^\circ\) (given perpendiculars). 4. **Write the congruence statement:** - \(\triangle ABC \cong \triangle ADC\) by ASA since two angles and the included side are equal. 5. **Summary:** - \(\angle BAC = \angle DAC\) (given, bisector). - \(AC = AC\) (common side). - \(\angle ABC = \angle ADC = 90^\circ\) (given, perpendiculars). - Therefore, \(\triangle ABC \cong \triangle ADC\) by ASA. This completes the proof.