Question

$$ \overleftrightarrow { f h } $$ bisects $$ \angle g h i $$ and $$ \angle g f i $$. complete the proof that $$ \triangle f g h \cong \triangle f i h $$.

1 $$ \overleftrightarrow { f h } $$ bisects $$ \angle g h i $$ given
2 $$ \overleftrightarrow { f h } $$ bisects $$ \angle g f i $$ given
3 $$ \angle f h g \cong \angle f h i $$
4 $$ \angle g f h \cong \angle h f i $$
5 $$ \overline { f h } \cong \overline { f h } $$ reflexive property of congruence
6 $$ \triangle f g h \cong \triangle f i h $$

Explanation:

Step 1: Analyze Statement 4

To determine the reason for \( \angle GFH \cong \angle HFI \), we recall the definition of an angle bisector. An angle bisector divides an angle into two congruent angles. Since \( \overrightarrow{FH} \) bisects \( \angle GFI \) (from Statement 2), it splits \( \angle GFI \) into \( \angle GFH \) and \( \angle HFI \), so these two angles must be congruent by the definition of an angle bisector.

Step 2: Analyze Statement 5

For \( \overline{FH} \cong \overline{FH} \), the reason is the Reflexive Property of Congruence, which states that any segment is congruent to itself.

Step 3: Analyze Statement 6

To prove \( \triangle FGH \cong \triangle FIH \), we use the Angle - Angle - Side (AAS) Congruence Criterion. We have \( \angle FHG \cong \angle FHI \) (Statement 3), \( \angle GFH \cong \angle HFI \) (Statement 4), and \( \overline{FH} \cong \overline{FH} \) (Statement 5). So the reason for \( \triangle FGH \cong \triangle FIH \) is the AAS Congruence Theorem.

Filling in the Reasons:

• Statement 4 Reason: Definition of Angle Bisector (Since \( \overrightarrow{FH} \) bisects \( \angle GFI \), it creates two congruent angles \( \angle GFH \) and \( \angle HFI \))
• Statement 5 Reason: Reflexive Property of Congruence (A segment is congruent to itself)
• Statement 6 Reason: AAS (Angle - Angle - Side) Congruence Theorem (We have two pairs of congruent angles and a non - included congruent side)

Answer:

• Statement 4 Reason: Definition of Angle Bisector
• Statement 5 Reason: Reflexive Property of Congruence
• Statement 6 Reason: AAS Congruence Theorem