We have proven that ∠BEC ≅ ∠BCE by using the alternate interior angles theorem
Proof of angles
From the question, we are to write a proof to prove that ∠BEC ≅ ∠BCE
From the given information, we have that
BC || EF
Note that line DE is a transversal
Thus, by the alternate interior angles theorem, we have that
∠BCE ≅ ∠FEC
Also, we were given that line EC bisects ∠BEF
That is,
Line EC divides ∠BEF into two equal angles.
Therefore,
∠BEC ≅ ∠FEC
By the substitution property of equality, we get
∠BEC ≅ ∠BCE
Hence, we have proven that ∠BEC ≅ ∠BCE by using the alternate interior angles theorem
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