Question

∠gfk ≅ ∠hij, gh ≅ jk, and ∠fkg ≅ ∠ihj. complete the proof that hi ≅ fk.

statement
1 ∠gfk ≅ ∠hij
2 (overline{gh}) ≅ (overline{jk})
3 ∠fkg ≅ ∠ihj
4 gk = jk + gj
5 hj = gh + gj
6 gk = gh + gj
7 hj = gk
8 △fgk ≅ △ijh
9 (overline{hi}) ≅ (overline{fk})

reason
given
given
given

substitution
transitive property of equality
aas
cpctc

Explanation:

Step1: Segment Addition Postulate

Segments can be added. So, \( GK = JK + GJ \) (Segment Addition Postulate) and \( HJ=GH + GJ \) (Segment Addition Postulate).

Step2: Substitution

Since \( \overline{GH}\cong\overline{JK}\), then \( GH = JK \). Substitute \( JK\) with \( GH\) in \( GK = JK + GJ\), we get \( GK=GH + GJ\).

Step3: Transitive Property of Equality

Because \( HJ = GH + GJ\) and \( GK=GH + GJ\), by Transitive Property of Equality, \( HJ = GK\).

Step4: Triangle Congruence (AAS)

We have \( \angle GFK\cong\angle HIJ\), \( \angle FKG\cong\angle IHJ\), and \( HJ = GK\). So, \( \triangle FGK\cong\triangle IJH\) (AAS: Angle - Angle - Side).

Step5: CPCTC

Since \( \triangle FGK\cong\triangle IJH\), by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), \( \overline{HI}\cong\overline{FK}\).

Answer:

The reasons for statements 4 - 9 are: 4. Segment Addition Postulate; 5. Segment Addition Postulate; 6. Substitution (because \( \overline{GH}\cong\overline{JK}\)); 8. AAS (Angle - Angle - Side); 9. CPCTC (Corresponding Parts of Congruent Triangles are Congruent).