Question

i is the midpoint of \\( \overline { f h } \\) and \\( \overline { e g } \\). complete the proof that \\( \triangle e h i \cong \triangle g f i \\).

1 i is the midpoint of \\( \overline { f h } \\)
2 i is the midpoint of \\( \overline { e g } \\)
3 \\( \overline { f i } \cong \overline { h i } \\)
4 \\( \overline { e i } \cong \overline { g i } \\)
5 \\( \angle e i h \cong \angle f i g \\)
6 \\( \triangle e h i \cong \triangle g f i \\)

Explanation:

Step1: Definition of mid - point

Since \(I\) is the mid - point of \(\overline{FH}\), by the definition of a mid - point (a mid - point divides a line segment into two equal parts), we have \(\overline{FI}\cong\overline{HI}\).

Step2: Definition of mid - point

Since \(I\) is the mid - point of \(\overline{EG}\), by the definition of a mid - point, we have \(\overline{EI}\cong\overline{GI}\).

Step3: Vertical angles

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

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

In \(\triangle EHI\) and \(\triangle GFI\), we have \(\overline{EI}\cong\overline{GI}\) (from step 2), \(\angle EIH\cong\angle FIG\) (from step 3), and \(\overline{HI}\cong\overline{FI}\) (from step 1). By the SAS (Side - Angle - Side) congruence criterion (if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent), \(\triangle EHI\cong\triangle GFI\).

Answer:

1. Given
2. Given
3. Definition of mid - point
4. Definition of mid - point
5. Vertical angles theorem
6. SAS (Side - Angle - Side) congruence criterion