Question

given: $\triangle afc cong \triangle efc$, $overline{bc} cong overline{dc}$ and $angle ghf cong angle gif$.
prove: $\triangle ahb cong \triangle eid$.

step\tstatement\treason
1\t$\triangle afc cong \triangle efc$
$overline{bc} cong overline{dc}$
$angle ghf cong angle gif$\tgiven
2\t$overline{ac} cong overline{ec}$\tcorresponding parts of congruent triangles are congruent (cpctc)
3\t$overline{ab} cong overline{ed}$\tcongruent segments subtracted from congruent segments form congruent segments
4\tblank for statement\t
5\t$angle ahb cong angle ghf$\tvertical angles are congruent
6\t$angle eid cong angle gif$\tvertical angles are congruent
7\t$angle ahb cong angle eid$\ttransitive property
8\t$\triangle ahb cong \triangle eid$\taas

Explanation:

Step1: Identify missing congruent angles

From $\triangle AFC \cong \triangle EFC$, by CPCTC, $\angle A \cong \angle E$.

Step2: Fill the missing statement

The missing statement for Step 4 is $\boldsymbol{\angle A \cong \angle E}$ with the reason "Corresponding Parts of Congruent Triangles are Congruent (CPCTC)".

Step3: Verify full proof flow

1. Use given congruent triangles to get $\overline{AC} \cong \overline{EC}$ (CPCTC)
2. Subtract congruent segments $\overline{BC} \cong \overline{DC}$ to get $\overline{AB} \cong \overline{ED}$
3. Get $\angle A \cong \angle E$ (CPCTC from $\triangle AFC \cong \triangle EFC$)
4. Use vertical angles to link $\angle AHB \cong \angle GHF$ and $\angle EID \cong \angle GIF$
5. Use transitive property to get $\angle AHB \cong \angle EID$
6. Prove $\triangle AHB \cong \triangle EID$ via AAS

Answer:

The missing statement for Step 4 is $\boldsymbol{\angle A \cong \angle E}$ with the reason Corresponding Parts of Congruent Triangles are Congruent (CPCTC).