Question

$$\overline{hi}\parallel\overline{fg},\overline{gi}\cong\overline{fh},$$ and $$\angle gfh\cong\angle fgi$$. complete the proof that $$\triangle ghi\cong\triangle fih$$.

	statement	reason
2	$$\overline{gi}\cong\overline{fh}$$	given
3	$$\angle gfh\cong\angle fgi$$	given
4	$$\angle gfh\cong\angle fhi$$	alternate interior angles theorem
5	$$\angle gih\cong\angle fgi$$	asa
6	$$\angle fgi\cong\angle fhi$$	transitive property of equality
7	$$\angle gih\cong\angle fhi$$	aas<br>additive property of length<br>all right angles are congruent<br>alternate interior angles theorem<br>asa
8	$$\overline{hi}\cong\overline{hi}$$
9	$$\triangle ghi\cong\triangle fih$$

Explanation:

Step 1: Alternate Interior Angles Theorem

Since \( \overline{HI}\parallel\overline{FG}\), by the Alternate Interior Angles Theorem, \( \angle GFH\cong\angle FHI\) (Statement 4).

Step 2: Transitive Property of Congruence

Given \( \angle GFH\cong\angle FGI\) (Statement 3) and \( \angle GFH\cong\angle FHI\) (Statement 4), by the Transitive Property of Congruence, \( \angle FGI\cong\angle FHI\) (Statement 6).

Step 3: Reflexive Property

For any segment \( \overline{HI}\), \( \overline{HI}\cong\overline{HI}\) by the Reflexive Property (Statement 8).

Step 4: AAS (Angle - Angle - Side) Congruence Criterion

We have \( \overline{GI}\cong\overline{FH}\) (Statement 2), \( \angle GIH\cong\angle FHI\) (Statement 7, derived from previous angle - congruence relations), and \( \overline{HI}\cong\overline{HI}\) (Statement 8). By the AAS (Angle - Angle - Side) congruence criterion, \( \triangle GHI\cong\triangle FIH\) (Statement 9).

Answer:

The reason for Statement 6 is the Transitive Property of Congruence, and the reason for Statement 9 is AAS (Angle - Angle - Side).