The prove, ∠ BCF ≅ ∠ DCF. is achieved using Angle addition postulate
How to complete the proof
The two column proof is written as follows
Statement Reason
AE ⊥ FC given
∠ A ≅ ∠ E given
∠ B ≅ ∠ D given
AC ≅ CE definition of midpoint
Δ BAC ≅ Δ DEC AAS congruence theorem
∠ BCA ≅ ∠ DCE CPCTC
∠ ACF ≅ ∠ DCF right angles are equal
∠ ACF = ∠ BCA +∠ BCF Angle addition postulate
∠ DCF = ∠ DCE +∠ DCF Angle addition postulate
∠ DCE +∠ DCF = ∠ BCA +∠ BCF Substitution
∠ BCA +∠ DCF = ∠ BCA +∠ BCF Substitution
∠ DCF = ∠ BCF Simplifying