Point D is taken inside an undeveloped corner A. From this point, perpendiculars are drawn

Point D is taken inside an undeveloped corner A. From this point, perpendiculars are drawn to the sides of the corner, angle ADB = angle ADC. Prove that ray AD is the bisector of angle A.

The perpendiculars DB and DC form two right-angled triangles ADB and ADC, which, by condition, have an angle ADB = ADC. The hypotenuse AD is common for both triangles, then the triangles ADB and ADC are equal according to the third sign of equality of right-angled triangles, according to the hypotenuse and acute angle. Then the angle BAD = CAD, and therefore AD is the bisector of the angle BAC, which was required to be proved.