Prove that triangle MOK is isosceles if you know that angle 1 is equal to angle 2.

Given:
triangle MОК,
angle ОМК = angle ОМК.
Prove that triangle MOK is isosceles.
Proof:
1) Consider the IOC triangle. Let’s draw the OP – the perpendicular from the top O to the side of the MK;
2) Consider two right-angled triangles MOP and OPK. They have a leg OP – a common one and an angle ОМК = angle ОМК. Therefore, according to the leg and the acute angle, the triangle MOP = triangle OPC. Then MO = OK. This means that the MOC triangle is isosceles. Q.E.D.