1. **State the problem:** Given points and angles with various congruences and perpendicularities, prove relationships between angles and triangles using geometric theorems. 2. **Given:** - C is midpoint of AE, so $AC \cong CE$. - $AE \perp FC$, so $\angle ECF$ is a right angle. - $\angle A \cong \angle E$, $\angle B \cong \angle D$. - $\triangle ABC \cong \triangle EDC$ by AAS (Angle-Angle-Side). - $\angle ACB \cong \angle ECD$. - $\angle ACB$ and $\angle BCF$ are complementary. - $\angle ECD$ and $\angle DCF$ are complementary. - $\angle BCF \cong \angle DCF$. 3. **Use midpoint definition:** $$AC = CE$$ 4. **Use perpendicularity:** Since $AE \perp FC$, $\angle ECF$ is a right angle, so $$\angle ECF = 90^\circ$$ 5. **Complementary angles:** If two angles form a right angle, they are complementary: $$\angle ACB + \angle BCF = 90^\circ$$ $$\angle ECD + \angle DCF = 90^\circ$$ 6. **Given $\angle BCF \cong \angle DCF$ and both complementary to $\angle ACB$ and $\angle ECD$ respectively, by the property that if two angles are complements of the same angle (or congruent angles), then they are congruent, we have:** $$\angle ACB \cong \angle ECD$$ 7. **Triangle congruence:** Given $\triangle ABC \cong \triangle EDC$ by AAS, corresponding parts of congruent triangles are congruent (CPCTC), confirming all stated angle congruences. **Final conclusion:** The given conditions and theorems prove the congruences and complementary relationships among the angles and triangles as stated.