Question

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: △afc ≅ △efc, \overline{bc} ≅ \overline{dc} and ∠ghf ≅ ∠gif.
prove: △ahb ≅ △eid.
step
statement
reason
1
△afc ≅ △efc
\overline{bc} ≅ \overline{dc}
∠ghf ≅ ∠gif
given
2
∠ahb ≅ ∠eid
transitive property
3
△ahb ≅ △eid
aas
4
\overline{ac} ≅ \overline{ec}
corresponding parts of congruent triangles are congruent (cpctc)
5
∠a ≅ ∠e
corresponding parts of congruent triangles are congruent (cpctc)
6
∠ahb ≅ ∠ghf
vertical angles are congruent
7
∠eid ≅ ∠gif
vertical angles are congruent
8
\overline{ab} ≅ \overline{ed}
congruent segments subtracted from congruent segments form congruent segments

Explanation:

Step1: List given information

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

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

$\overline{AC}\cong\overline{EC}$, $\angle A\cong\angle E$ (Corresponding Parts of Congruent Triangles are Congruent (CPCTC))

Step3: Use congruent segments subtraction

$\overline{AB}\cong\overline{ED}$ (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

$\angle AHB\cong\angle EID$ (Transitive Property)

Step6: Prove triangle congruence

$\triangle AHB\cong\triangle EID$ (AAS)

Answer:

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