A straight line perpendicular to the bisector was drawn through the point M, which belongs to the bisector of the angle

A straight line perpendicular to the bisector was drawn through the point M, which belongs to the bisector of the angle with the vertex at the point O. This line intersects the sides of this angle at points A and B. Prove that triangle ABK = triangle ADK.

Since the straight line AB, by condition, is perpendicular to the bisector OM, the triangles AOM and BOM are rectangular.

Angle AOM = BOM, since OM is the bisector of angle AOB.

The OM side is common for both triangles.

Then the triangle AOM = BOM along the leg and the adjacent acute angle, according to the second sign of equality of right-angled triangles. Q.E.D.