Question

$\angle gfi \cong \angle hfi$ and $\overline{fi} \perp \overline{gh}$. complete the proof that $\triangle fhi \cong \triangle fgi$.

statement	reason
2 $\overline{fi} \perp \overline{gh}$	given
3 $\angle fig \cong \angle fih$	all right angles are congruent
4 $\overline{fi} \cong \overline{fi}$
5 $\triangle fhi \cong \triangle fgi$

Explanation:

Step1: Identify common segment

$\overline{FI}$ is a side shared by both $\triangle FHI$ and $\triangle FGI$. By the reflexive property, a segment is congruent to itself, so $\overline{FI} \cong \overline{FI}$.

Step2: Match congruence criteria

We have:

1. $\angle GFI \cong \angle HFI$ (Given)
2. $\overline{FI} \cong \overline{FI}$ (Reflexive Property)
3. $\angle FIG \cong \angle FIH$ (Proven right angles congruent)

This matches the ASA congruence postulate: two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, so the triangles are congruent.

Answer:

1. Reason for $\overline{FI} \cong \overline{FI}$: Reflexive Property of Congruence
2. Reason for $\triangle FHI \cong \triangle FGI$: Angle-Side-Angle (ASA) Congruence Postulate