Question

∠fgk ≅ ∠hji, overline{gh} ≅ overline{jk}, and ∠fkg ≅ ∠ihj. complete the proof that △hji ≅ △kfg.
statement\treason
1 ∠fgk ≅ ∠hji\tgiven
2 overline{gh} ≅ overline{jk}\tgiven
3 ∠fkg ≅ ∠ihj\tgiven
4 gk = jk + gj\tadditive property of length
5 hj = gh + gj\t
6 gk = gh + gj\tsubstitution
7 hj = gk\ttransitive property of equality

Explanation:

Step1: Identify given congruences

$\angle FGK \cong \angle HJI$, $\overline{GH} \cong \overline{JK}$, $\angle FKG \cong \angle IHJ$

Step2: Apply segment addition

$\overline{GK} = \overline{JK} + \overline{GJ}$, $\overline{HJ} = \overline{GH} + \overline{GJ}$

Step3: Substitute congruent segments

Since $\overline{GH} \cong \overline{JK}$, substitute into $\overline{HJ} = \overline{GH} + \overline{GJ}$ to get $\overline{HJ} = \overline{JK} + \overline{GJ}$

Step4: Use transitive equality

From $\overline{GK} = \overline{JK} + \overline{GJ}$ and $\overline{HJ} = \overline{JK} + \overline{GJ}$, we get $\overline{HJ} = \overline{GK}$

Step5: Match ASA congruence

We have $\angle HJI \cong \angle FGK$, $\overline{HJ} \cong \overline{GK}$, $\angle IHJ \cong \angle FKG$, so $\triangle HIJ \cong \triangle KFG$ by ASA

Answer:

Statement	Reason
2. $\overline{GH} \cong \overline{JK}$	Given
3. $\angle FKG \cong \angle IHJ$	Given
4. $GK = JK + GJ$	Additive Property of Length
5. $HJ = GH + GJ$	Additive Property of Length
6. $GK = GH + GJ$	Substitution
7. $HJ = GK$	Transitive Property of Equality
8. $\triangle HIJ \cong \triangle KFG$	ASA

Final proof completion: The missing reason for Statement 5 is Additive Property of Length, and the final congruence $\triangle HIJ \cong \triangle KFG$ is proven by ASA.