Question

δegi is equilateral. complete the proof that \\(\overline{ef} \cong \overline{hi}\\).

(image of a geometric figure with points f, h, e, i, g, and a table with statements and reasons:

1. δegi is equilateral - given
2. ∠f ≅ ∠h - given
3. ∠hgi ≅ ∠egf - vertical angle theorem
4. \\(\overline{gi} \cong \overline{eg}\\) - definition of equilateral triangle
5. δghi ≅ δgfe
6. \\(\overline{ef} \cong \overline{hi}\\))

Explanation:

Step1: Identify given info

$\triangle EGI$ is equilateral (given), $\angle F \cong \angle H$ (given), $\angle HGI \cong \angle EGF$ (vertical angles).

Step2: Equilateral triangle sides

$\overline{GI} \cong \overline{EG}$ (def of equilateral triangle).

Step3: Prove triangle congruence

$\triangle GHI \cong \triangle GFE$ by AAS (2 angles + non-included side).

Step4: Corresponding sides congruent

$\overline{EF} \cong \overline{HI}$ (CPCTC).

Answer:

$\overline{EF} \cong \overline{HI}$ is proven via AAS congruence and CPCTC.