Segments AB and KP intersect at point O so that AO = OB, OK = OP. Prove that AOP = BOK.

In fact, this is not difficult to prove.

We will now have to prove the equality of 2 triangles.

And as you know, triangles are equal in 3 elements.

In the condition, we are already given 2 equal elements.

3 equal elements of these triangles will be vertical corners. Since, by the property of vertical angles, the vertical angles are equal.

From here we write out all equal parts:

1) angle KOВ = angle AOP (since they are vertical).
2) KO = OP (by condition).
3) OB = AO (by condition).

And it turns out that the triangles are equal in 2 sides and the angle between them.