Question

gh ≅ fi and fg ≅ hi. complete the proof that ∠g ≅ ∠i. image of triangle with points g, f, h, i and a table with statements and reasons

Explanation:

Step1: Identify Given Information

We know $\overline{GH} \cong \overline{FI}$ (given), $\overline{FG} \cong \overline{HI}$ (given), and we need to consider $\overline{FH}$ which is common to both $\triangle FGH$ and $\triangle HIF$. So $\overline{FH} \cong \overline{FH}$ by the Reflexive Property of Congruence.

Step2: Apply SSS Congruence

For $\triangle FGH$ and $\triangle HIF$:

• $\overline{GH} \cong \overline{FI}$ (given)
• $\overline{FG} \cong \overline{HI}$ (given)
• $\overline{FH} \cong \overline{FH}$ (Reflexive Property)

By the Side - Side - Side (SSS) Congruence Postulate, $\triangle FGH \cong \triangle HIF$.

Step3: Corresponding Parts of Congruent Triangles

Since $\triangle FGH \cong \triangle HIF$, their corresponding angles are congruent. $\angle G$ and $\angle I$ are corresponding angles, so $\angle G \cong \angle I$ by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem.

Now, let's fill in the reasons:

1. $\overline{GH} \cong \overline{FI}$: Given
2. $\overline{FG} \cong \overline{HI}$: Given
3. $\overline{FH} \cong \overline{FH}$: Reflexive Property of Congruence
4. $\triangle FGH \cong \triangle HIF$: SSS Congruence Postulate
5. $\angle G \cong \angle I$: CPCTC

Answer:

1. Reason: Given
2. Reason: Given
3. Reason: Reflexive Property of Congruence
4. Reason: SSS Congruence Postulate
5. Reason: CPCTC