Question

reorder the steps of the proof to make sure that steps that are logically dependent on prior steps are in the proper order.
given: \\( \triangle a f c \cong \triangle e f c, \overline{b c} \cong \overline{d c} \\) and \\( \angle g h f \cong \angle g i f \\).
prove: \\( \triangle a h b \cong \triangle e i d \\).
\\( \begin{array} { l l l } { \text { step } } & { \text { statement } } & { \text { reason } } \\ { 1 } & { \begin{array} { l } { \triangle a f c cong \triangle e f c } \\ { overline { b c } cong overline { d c } } \\ { angle g h f cong angle g i f } end{array} } & { \text { given } } \\ { 2 } & { angle a h b cong angle e i d } & { \text { transitive property } } \\ { 3 } & { \triangle a h b cong \triangle e i d } & { \text { aas } } \\ { 4 } & { overline { a c } cong overline { e c } } & { \text { corresponding parts of congruent triangles are congruent (cpctc) } } \\ { 5 } & { angle a cong angle e } & { \text { corresponding parts of congruent triangles are congruent (cpctc) } } \\ { 6 } & { angle a h b cong angle g h f } & { \text { vertical angles are congruent } } \\ { 7 } & { angle e i d cong angle g i f } & { \text { vertical angles are congruent } } \\ { 8 } & { overline { a b } cong overline { e d } } & { \text { congruent segments subtracted from congruent segments form congruent segments } } end{array} \\)

Explanation:

Step1: State given information

$\triangle AFC\cong\triangle EFC$, $\overline{BC}\cong\overline{DC}$, $\angle GHF\cong\angle GIF$

Step2: Use CPCTC on $\triangle AFC\cong\triangle EFC$

$\angle A\cong\angle E$, $\overline{AC}\cong\overline{EC}$

Step3: Use segment subtraction

Since $\overline{AC}\cong\overline{EC}$ and $\overline{BC}\cong\overline{DC}$, then $\overline{AB}\cong\overline{ED}$ (Congruent segments subtracted from congruent segments form congruent segments)

Step4: Use vertical angles congruence

$\angle AHB\cong\angle GHF$, $\angle EID\cong\angle GIF$ (Vertical angles are congruent)

Step5: Use transitive property on angles

Since $\angle GHF\cong\angle GIF$, $\angle AHB\cong\angle EID$ (Transitive Property)

Step6: Prove triangle congruence

In $\triangle AHB$ and $\triangle EID$, $\angle A\cong\angle E$, $\overline{AB}\cong\overline{ED}$, $\angle AHB\cong\angle EID$. So, $\triangle AHB\cong\triangle EID$ (AAS)

Answer:

1. $\triangle AFC\cong\triangle EFC$, $\overline{BC}\cong\overline{DC}$, $\angle GHF\cong\angle GIF$ (Given)
2. $\angle A\cong\angle E$ (CPCTC)
3. $\overline{AC}\cong\overline{EC}$ (CPCTC)
4. $\overline{AB}\cong\overline{ED}$ (Congruent segments subtracted from congruent segments form congruent segments)
5. $\angle AHB\cong\angle GHF$ (Vertical angles are congruent)
6. $\angle EID\cong\angle GIF$ (Vertical angles are congruent)
7. $\angle AHB\cong\angle EID$ (Transitive Property)
8. $\triangle AHB\cong\triangle EID$ (AAS)