Question

$$ \overline { h i } \parallel \overline { f g }, \overline { g i } \cong \overline { f h }, $$ and $$ \angle g f h \cong \angle f g i $$. complete the proof that $$ \triangle g h i \cong \triangle f h $$.

1 $$ \overline { h i } \parallel \overline { f g } $$
2 $$ \overline { g i } \cong \overline { f h } $$
3 $$ \angle g f h \cong \angle f g i $$
4 $$ \angle g f h \cong \angle f h i $$
5 $$ \angle g i h \cong \angle f g i $$
6 $$ \angle f g i \cong \angle f h i $$
7 $$ \angle g i h \cong \angle f h i $$
8 $$ \overline { h i } \cong \overline { h i } $$
9 $$ \triangle g h i \cong \triangle f h $$

Explanation:

Step1: Given

Statements 1 - 3 are given in the problem.

Step2: Alternate interior angles

Since \( \overline{HI}\parallel\overline{FG}\), by the alternate - interior angles theorem, \( \angle GFH\cong\angle FHI\) (statement 4).

Step3: Transitive property

From statement 3 (\( \angle GFH\cong\angle FGI\)) and statement 4 (\( \angle GFH\cong\angle FHI\)), by the transitive property of congruence, \( \angle FGI\cong\angle FHI\) (statement 6).

Step4: Reflexive property

For statement 8, \( \overline{HI}\cong\overline{HI}\) by the reflexive property of congruence.

Step5: ASA (Angle - Side - Angle)

In \( \triangle GHI\) and \( \triangle FIH\), we have \( \overline{GI}\cong\overline{FH}\) (statement 2), \( \angle GIH\cong\angle FHI\) (statement 7), and \( \overline{HI}\cong\overline{HI}\) (statement 8). By the ASA (Angle - Side - Angle) congruence criterion, \( \triangle GHI\cong\triangle FIH\) (statement 9).

Answer:

1. Given
2. Given
3. Given
4. Alternate - interior angles theorem (\( \overline{HI}\parallel\overline{FG}\))
5. (This row may be a mis - step in the original proof structure. If we assume it's a typo and we are building up to the ASA criterion)
6. Transitive property (from statements 3 and 4)
7. (If we assume statement 5 was a wrong label and statement 7 is derived from previous angle congruences in the correct order)
8. Reflexive property of congruence
9. ASA (Angle - Side - Angle) congruence criterion (\( \overline{GI}\cong\overline{FH}\), \( \angle GIH\cong\angle FHI\), \( \overline{HI}\cong\overline{HI}\))