Lines AB and CD intersect at point O from which equal segments AO and OB are laid on line AB

Lines AB and CD intersect at point O from which equal segments AO and OB are laid on line AB and equal segments OC and OD are laid on line CD. Prove that AD = BC.

Given:

AO = OB,

OC = OD,

Prove:

AD = BC,

I. Consider triangles AOD and BOC:

1) angle 1 = angle 2 – vertical angles,

2) CO = OD – by condition,

3) AO = OB – by condition,

It follows that the triangles AOD and BOC are similar.

II. Since triangles AOD and BOC are similar, AD = BC.