Question

complete the proof that $\triangle fgi\cong\triangle ehi$.

Explanation:

Step1: Given

From the figure, we can directly observe that \(\overline{FI}\cong\overline{EI}\) (given as marked segments).

Step2: Given

Also, from the figure, \(\overline{HI}\cong\overline{GI}\) (given as marked segments).

Step3: Vertical angles are congruent

\(\angle EIH\) and \(\angle FIG\) are vertical angles. By the vertical - angles theorem, \(\angle EIH\cong\angle FIG\).

Step4: SAS (Side - Angle - Side) congruence criterion

In \(\triangle FGI\) and \(\triangle EHI\), we have \(\overline{FI}\cong\overline{EI}\), \(\angle FIG\cong\angle EIH\), and \(\overline{GI}\cong\overline{HI}\). By the Side - Angle - Side (SAS) congruence criterion, \(\triangle FGI\cong\triangle EHI\).

Answer:

1. Given
2. Given
3. Vertical angles are congruent
4. SAS (Side - Angle - Side)