Math Snap

STEP 1

1. $\overline{D E} \cong \overline{C E}$
2. $\overline{F E}$ bisects $\angle D E C$
3. The diagram accurately represents the given information
4. Points A and B are the intersections of $\overline{FD}$ and $\overline{FC}$ with $\overline{DC}$ respectively

STEP 2

1. Analyze the given information
2. Identify congruent triangles
3. Prove that $\overline{FA} \cong \overline{FB}$

STEP 3

Analyze the given information:
$\overline{D E} \cong \overline{C E}$ implies that E is the midpoint of $\overline{DC}$.
$\overline{F E}$ bisects $\angle D E C$, so $\angle DEF \cong \angle CEF$.

STEP 4

Identify congruent triangles:
We can prove that $\triangle DEF \cong \triangle CEF$ using the SAS (Side-Angle-Side) congruence criterion:
1. $\overline{D E} \cong \overline{C E}$ (given)
2. $\angle DEF \cong \angle CEF$ ($\overline{F E}$ bisects $\angle D E C$)
3. $\overline{EF}$ is common to both triangles
Therefore, $\triangle DEF \cong \triangle CEF$

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 $\overline{FA} \cong \overline{FB}$.