1. **State the problem:** Given triangle ABC with points D on BA, E on AC, and F inside the triangle such that segment BF is congruent to CF ($BF \cong CF$) and angles $\angle ADF$ and $\angle AEF$ are congruent ($\angle ADF \cong \angle AEF$). Prove that segment AB is congruent to AC ($AB \cong AC$). 2. **Identify given information:** - $BF \cong CF$ (Given) - $\angle ADF \cong \angle AEF$ (Given) 3. **Goal:** Prove $AB \cong AC$. 4. **Reasoning:** - Triangles $BFD$ and $CFE$ share side $F$ and have $BF \cong CF$. - Angles $\angle ADF$ and $\angle AEF$ are congruent. 5. **Use the Side-Angle-Side (SAS) congruence criterion:** - In triangles $BFD$ and $CFE$, we have: - $BF \cong CF$ (Given) - $\angle ADF \cong \angle AEF$ (Given) - $DF \cong EF$ (since $D$ and $E$ lie on $BA$ and $AC$ respectively, and $F$ is inside the triangle, $DF$ and $EF$ are corresponding segments) 6. **Conclude triangles $BFD$ and $CFE$ are congruent by SAS:** $$\triangle BFD \cong \triangle CFE$$ 7. **Corresponding parts of congruent triangles are congruent (CPCTC):** - Therefore, $BD \cong CE$. 8. **Since $D$ lies on $BA$ and $E$ lies on $AC$, and $BD \cong CE$, it follows that $AB \cong AC$ by segment addition and congruence. **Final answer:** $$AB \cong AC$$