Prove that you have constructed point C on segment EF such that angle ACE is congruent to angle BCF. (Points A and B are on the same side of segment EF, but have different distances to the segment.)

Question

I am not sure if I am on the right track but this is how I have tried prove it: I have proven a new angle, angle ICF, is congruent to angle ACE by the Vertical Angle Theorem (after introducing line AC and point I at the intersect of line AC and line BD (which is perpendicular to segment EF)). I have introduced point J on line BD such that JD=ID and proved triangle CJD is congruent to triangle CID by SAS (CD=CD by Reflexive Property, angle JDC is congruent to angle IDC because line BD is perpendicular to segment EF (which contains point C) therefore angles JDC and IDC are right angles and all right angles are congruent, and I had introduced point J so that JD=ID). At the moment I cannot see how I would prove point J is point B, so I am not sure if the above reasoning is in the right direction.

Thank you for your help!