By which reason can it be proven that triangles dab and dac are congruent

ASA

Step-by-step explanation:

You are given 2 equal sets of angles.

<ABD and <ACD    (right angles)

<BAD and <CDA    (bisected angle)

You can get <ADB = <ADC because every triangle has 180 degrees. You know two of the angles: you can show that  the 3 rd set are equal a subtraction postulate.

AD = AD              (Reflexive property)

The triangles are congruent by ASA

If AAS is acceptable then it is correct in this case.

AD = AD

The two right angles are equal

The two angles created by the angle bisector are equal.

These three facts give AAS, but please see my comment below.