Suppose O lies on the interior of <ABC, D lies on the interior of <ABO and E lies on the interior of < OBC.  

It is also given that m<DBO = m<EBC and m<ABC = 120, m<DBC = 90.

What is the relationships between <DBE and <OBC ?

From the information in the problem we obtain the following drawing.

Since m<DBO = m<EBC and <OBE is part of both <DBO and <OBE, it follows that

m<DBE = m<OBC.

Notice this in the following diagram

Since m<ABC = 120 and m<DBC = 90, m<ABD = 30.

We know this from the angle addition postulate, m<ABD + m<DBC = m<ABC.