Question

i is the midpoint of \\( \overline { f h } \\) and \\( \overline { e g } \\) and \\( \overline { e h } \cong \overline { f g } \\). complete the proof that \\( \triangle f g i \cong \triangle h e i \\).
1 i is the midpoint of \\( \overline { f h } \\) given
2 i is the midpoint of \\( \overline { e g } \\) given
3 \\( \overline { e h } \cong \overline { f g } \\) given
4 \\( \overline { f i } \cong \overline { h i } \\) definition of midpoint
5 \\( \overline { e i } \cong \overline { g i } \\) definition of midpoint
6

Explanation:

Step1: Identify vertical angles

$\angle FIG\cong\angle HIE$ (Vertical angles are congruent)

Step2: Apply SSS or SAS congruence

In $\triangle FGI$ and $\triangle HEI$:

• $\overline{FI}\cong\overline{HI}$ (From step 4, definition of mid - point)
• $\overline{EI}\cong\overline{GI}$ (From step 5, definition of mid - point)
• $\angle FIG\cong\angle HIE$ (Vertical angles)

By the SAS (Side - Angle - Side) congruence criterion, $\triangle FGI\cong\triangle HEI$

Answer:

$\triangle FGI\cong\triangle HEI$ (by SAS congruence)