Question

fj bisects ∠gik and ∠hfj. complete the proof that (overline{hi} cong overline{ij}).
figure of a geometric diagram with points j, k, f, i, g, h

	statement	reason
1	(overrightarrow{fj}) bisects ∠gik
2	(overrightarrow{fj}) bisects ∠hfj
3	∠jik ≅ ∠gih
4	∠fik ≅ ∠fig
5	∠ifj ≅ ∠hfj
6	(mangle fij = mangle fik + mangle jik)
7	(mangle fih = mangle fig + mangle gih)
8	(mangle fij = mangle fig + mangle gih)
9	(mangle fih = mangle fij)
10	(overline{fi} cong overline{fi})
11	(\triangle fij cong \triangle fih)
12	(overline{hi} cong overline{ij})

Explanation:

Step1: Recall Angle Bisector Definition

An angle bisector divides an angle into two congruent angles. So for step 1, since \( \overrightarrow{FI} \) bisects \( \angle GIK \), the reason is "Definition of Angle Bisector".

Step2: Apply Angle Bisector Definition Again

For step 2, \( \overrightarrow{FI} \) bisects \( \angle HFJ \), so the reason is also "Definition of Angle Bisector".

Step3: Use Angle Bisector Result

From step 1, \( \angle JIK \cong \angle GIH \) because the bisector creates two congruent angles. Reason: "Definition of Angle Bisector (from step 1)".

Step4: Use Angle Bisector Result

From step 2, \( \angle FIK \cong \angle FIG \) because the bisector creates two congruent angles. Reason: "Definition of Angle Bisector (from step 2)".

Step5: Vertical Angles or Bisector? Wait, \( \angle IFJ \) and \( \angle HFI \)? Wait, no, from step 2, \( \overrightarrow{FI} \) bisects \( \angle HFJ \), so \( \angle IFJ \cong \angle HFI \) by "Definition of Angle Bisector".

Step6: Angle Addition Postulate

The measure of an angle is the sum of its parts. So \( m\angle FIJ = m\angle FIK + m\angle JIK \) by "Angle Addition Postulate".

Step7: Angle Addition Postulate

Similarly, \( m\angle FIH = m\angle FIG + m\angle GIH \) by "Angle Addition Postulate".

Step8: Substitution (from steps 3 and 4)

Since \( \angle FIK \cong \angle FIG \) (step 4) and \( \angle JIK \cong \angle GIH \) (step 3), substitute into step 6: \( m\angle FIJ = m\angle FIG + m\angle GIH \), so \( m\angle FIJ = m\angle FIH \) (step 8 is stating \( m\angle FIJ = m\angle FIG + m\angle GIH \), which is substitution from steps 3,4,6). Wait, step 8: \( m\angle FIJ = m\angle FIG + m\angle GIH \) by substitution (from steps 3,4,6).

Step9: Transitive Property or Substitution

From steps 7 and 8, \( m\angle FIH = m\angle FIG + m\angle GIH = m\angle FIJ \), so \( m\angle FIH = m\angle FIJ \), hence \( \angle FIH \cong \angle FIJ \) (by definition of congruent angles, equal measures).

Step10: Reflexive Property

\( \overline{FI} \cong \overline{FI} \) by "Reflexive Property of Congruence".

Step11: ASA Congruence

We have \( \angle FIJ \cong \angle FIH \) (step 9), \( \overline{FI} \cong \overline{FI} \) (step 10), and \( \angle IFJ \cong \angle HFI \) (step 5). So \( \triangle FIJ \cong \triangle FIH \) by "ASA (Angle - Side - Angle) Congruence Postulate".

Step12: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \( \triangle FIJ \cong \triangle FIH \), then \( \overline{HI} \cong \overline{IJ} \) by "CPCTC".

Answer:

To complete the proof:

1. Reason: Definition of Angle Bisector
2. Reason: Definition of Angle Bisector
3. Reason: Definition of Angle Bisector (from 1, \( \overrightarrow{FI} \) bisects \( \angle GIK \))
4. Reason: Definition of Angle Bisector (from 2, \( \overrightarrow{FI} \) bisects \( \angle HFJ \))
5. Reason: Definition of Angle Bisector (from 2)
6. Reason: Angle Addition Postulate
7. Reason: Angle Addition Postulate
8. Reason: Substitution (from 3,4)
9. Reason: Substitution (from 7,8)
10. Reason: Reflexive Property of Congruence
11. Reason: ASA Congruence Postulate
12. Reason: CPCTC

Final answer for the proof steps follows the above reasoning, with \( \overline{HI} \cong \overline{IJ} \) proven via ASA and CPCTC.