Question

△fgi≅△hei. complete the proof that (overline{ef}congoverline{gh}).
1 △fgi≅△hei given
2 (overline{ei}congoverline{gi}) cpctc
3 (overline{fi}congoverline{hi}) cpctc
4 (∠eifcong∠gih)
5 △ghi≅△efi
6 (overline{ef}congoverline{gh}) cpctc

Explanation:

Step1: Vertical angles are congruent

$\angle EIF$ and $\angle GIH$ are vertical angles. By the vertical angles theorem, $\angle EIF\cong\angle GIH$.

Step2: SAS (Side - Angle - Side) congruence

We know that $\overline{EI}\cong\overline{GI}$, $\angle EIF\cong\angle GIH$, and $\overline{FI}\cong\overline{HI}$. So, by the SAS (Side - Angle - Side) congruence criterion, $\triangle GHI\cong\triangle EFI$.

Answer:

1. Vertical Angles Theorem; 5. SAS (Side - Angle - Side)