Suppose $ ABCD$ be the given quadrilateral and $ \angle ABC=\angle CDA$. Fix points $ A,B,C$. So $ D$ is on the line which is perpendicular to line $ AC$. Note that the point $ D$ for which $ \angle CDA=\angle ABC$ is unique. Note also that the point $ D$ for which $ ABCD$ is a kite satisfies $ \angle CDA=\angle ABC$. Hence $ ABCD$ must be a kite, which follows that $ AB+CD=BC+AD$. It is well-known that a circle can be inscribed in such quadrilateral.