Exercise 54 Page 382

Using the fact that AE and BD bisect each other at point C, we can conclude that BC ≅ CD and AC≅ CE.

We can also see that ∠ BCA and ∠ DCE are vertical angles. Therefore, m∠ BCA≅ m∠ DCE by the Vertical Angles Theorem.

Two sides and the included angle of △ BCA are congruent to two sides and the included angle of △ DCE. Thus, △ BCA≅ △ DCE by the Side-Angle-Side Theorem. To prove that DE ≅ DC, we need to prove that ∠ E ≅ ∠ DCE. Once we know this, DE ≅ DC because △ DCE is isosceles and the corresponding angles are congruent.

Because corresponding parts of corresponding triangles are congruent, If ∠ A ≅ ∠ BCA, then ∠ E ≅ ∠ DCE. Therefore, the additional statement ∠ A ≅ ∠ BCA is required. The correct choice is F.