Question

$\triangle ehi \cong \triangle fgi$. complete the proof that $\angle feh \cong \angle efg$.

(image of a quadrilateral with points f, g, h, e and intersection i of diagonals)

statement	reason
2 $\overline{eh} \cong \overline{fg}$
3 $\overline{ei} \cong \overline{fi}$
4 $\overline{gi} \cong \overline{hi}$
5 $\angle ehi \cong \angle fgi$
6 $eg = ei + gi$
7 $fh = fi + hi$
8 $eg = fi + hi$
9 $eg = fh$
10 $\triangle efh \cong \triangle feg$
11 $\angle feh \cong \angle efg$

Explanation:

Step1: Identify given congruence

Given $\Delta EHI \cong \Delta FGI$ (Premise)

Step2: Corresponding sides of congruent triangles

$\overline{EH} \cong \overline{FG}$ (CPCTC: Corresponding Parts of Congruent Triangles are Congruent)

Step3: Corresponding sides of congruent triangles

$\overline{EI} \cong \overline{FI}$ (CPCTC)

Step4: Corresponding sides of congruent triangles

$\overline{GI} \cong \overline{HI}$ (CPCTC)

Step5: Corresponding angles of congruent triangles

$\angle EHI \cong \angle FGI$ (CPCTC)

Step6: Segment addition postulate

$EG = EI + GI$ (Segment Addition Postulate)

Step7: Segment addition postulate

$FH = FI + HI$ (Segment Addition Postulate)

Step8: Substitute congruent segments

$EG = FI + HI$ (Substitution: $\overline{EI} \cong \overline{FI}$, $\overline{GI} \cong \overline{HI}$)

Step9: Substitute segment sums

$EG = FH$ (Substitution: $FH = FI + HI$)

Step10: Prove triangle congruence

$\Delta EFH \cong \Delta FEG$ (SSS: $\overline{EH} \cong \overline{FG}$, $\overline{EF} \cong \overline{FE}$ (reflexive), $EG = FH$)

Step11: Corresponding angles of congruent triangles

$\angle FEH \cong \angle EFG$ (CPCTC)

Answer:

The proof is completed with the reasons: 1. Given; 2. CPCTC; 3. CPCTC; 4. CPCTC; 5. CPCTC; 6. Segment Addition Postulate; 7. Segment Addition Postulate; 8. Substitution; 9. Substitution; 10. SSS; 11. CPCTC. Thus, $\angle FEH \cong \angle EFG$.