Question

fh bisects ∠gfi and ∠g ≅ ∠i. complete the proof that △fhi ≅ △fhg.

1. fh bisects ∠gfi
2. ∠g ≅ ∠i
3. ∠gfh ≅ ∠hfi
4. \overline{fh} ≅ \overline{fh}
5. △fhi ≅ △fhg

Explanation:

Step1: Given information

• \(FH\) bisects \(\angle GFI\) (Given)
• \(\angle G\cong\angle I\) (Given)

Step2: Angle - bisector property

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

Step3: Reflexive property

For any line segment \(FH\), \(\overline{FH}\cong\overline{FH}\) (Reflexive property of congruence)

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

In \(\triangle FHI\) and \(\triangle FHG\), we have \(\angle I\cong\angle G\) (Given), \(\angle HFI\cong\angle GFH\) (From angle - bisector), and \(\overline{FH}\cong\overline{FH}\) (Reflexive property). By the ASA congruence criterion, \(\triangle FHI\cong\triangle FHG\)

Answer:

1. Given
2. Given
3. Definition of angle bisector
4. Reflexive property of congruence
5. ASA (Angle - Side - Angle) congruence criterion