Equal segments ab and cd intersect at point O so that AO: OB = CO: OD = 2: 1. Prove that triangle AOD = triangle COB.

Given:
AB = CD,
point O – the point of intersection of segments AB and CD,
AO: ОВ = СО: ОD = 2: 1.
Prove that triangle AOD = triangle COB.
Evidence:
Consider triangle AOD and triangle COB. They have sides AO = CO and OD = OB because the ratio AO: OB = CO: OD = 2: 1 is fulfilled.
Angle AOD = angle of COB, since these angles are vertical, and the vertical angles are equal to each other. Therefore, the triangle AOD = the triangle of the COB on two sides and the angle between them. Q.E.D.