Question

fh bisects ∠gfi and ∠i ≅ ∠g. complete the proof that (overline{hi} cong overline{gh}).

Explanation:

Step1: Definition of angle bisector

Since \(\overrightarrow{FH}\) bisects \(\angle GFI\), by the definition of an angle bisector, \(\angle GFH\cong\angle HFI\).

Step2: Reflexive property

For any segment \(FH\), \(FH\cong FH\) by the reflexive property of congruence (a segment is congruent to itself).

Step3: AAS (Angle - Angle - Side) congruence criterion

We know that \(\angle I\cong\angle G\) (given), \(\angle GFH\cong\angle HFI\) (from the angle - bisector definition), and \(FH\cong FH\) (reflexive property). So, by the AAS (Angle - Angle - Side) congruence criterion, \(\triangle FGH\cong\triangle FIH\).

Step4: Corresponding parts of congruent triangles

If \(\triangle FGH\cong\triangle FIH\), then their corresponding parts are congruent. So, \(HI\cong GH\) (corresponding parts of congruent triangles are congruent).

Answer:

1. Definition of angle bisector
2. Given
3. Definition of angle bisector
4. Reflexive property of congruence
5. AAS (Angle - Angle - Side)
6. Corresponding parts of congruent triangles are congruent