QuestionGiven: $\overline{D E} \cong \overline{C E}$ and $\overline{F E}$ bisects $\angle D E C$. Prove: $\overline{F A} \cong \overline{F B}$.
Step Statement Reason
1 $\overline{D E} \cong \overline{C E}$ $\overline{F E}$ bisects $\angle D E C$ Given try Type of Statement

Studdy Solution
Prove that $\overline{FA} \cong \overline{FB}$:
3.1. Since $\triangle DEF \cong \triangle CEF$, we know that $\overline{FD} \cong \overline{FC}$
3.2. E is the midpoint of $\overline{DC}$, so $\overline{DE} \cong \overline{EC}$
3.3. $\triangle FDA \cong \triangle FCB$ by the SAS congruence criterion: - $\overline{FD} \cong \overline{FC}$ (from step 3.1) - $\angle FDA \cong \angle FCB$ (vertical angles) - $\overline{DA} \cong \overline{CB}$ (E is midpoint of $\overline{DC}$, so $\overline{DA} = \overline{DE}$ and $\overline{CB} = \overline{CE}$)
3.4. Since $\triangle FDA \cong \triangle FCB$, we can conclude that $\overline{FA} \cong \overline{FB}$
Therefore, we have proved that $\overline{FA} \cong \overline{FB}$.