Question

use the given information to prove that $\triangle abc \cong \triangle adc$.
given: $\angle acb$ is a right angle
$\angle acd$ is a right angle
$\angle bac \cong \angle dac$
prove: $\triangle abc \cong \triangle adc$

Explanation:

Step1: Identify Given Angles

$\angle ACB = 90^\circ$, $\angle ACD = 90^\circ$, so $\angle ACB \cong \angle ACD$ (right angles are congruent).

Step2: Identify Common Side

$AC$ is common to both $\triangle ABC$ and $\triangle ADC$, so $AC \cong AC$ (reflexive property).

Step3: Identify Given Angle Congruence

$\angle BAC \cong \angle DAC$ (given).

Step4: Apply AAS Congruence

In $\triangle ABC$ and $\triangle ADC$: $\angle BAC \cong \angle DAC$, $\angle ACB \cong \angle ACD$, $AC \cong AC$. By Angle - Angle - Side (AAS) congruence criterion, $\triangle ABC \cong \triangle ADC$.

Answer:

$\triangle ABC \cong \triangle ADC$ (proven by AAS congruence: $\angle BAC \cong \angle DAC$, $\angle ACB \cong \angle ACD$, $AC \cong AC$)