Question

$overline{fh} cong overline{fi}$, $angle j cong angle g$, and $angle gfh cong angle ifj$. complete the proof that $\triangle fgi cong \triangle fjh$.

(image of a geometric figure with points g, h, f, i, j and a table with statements and reasons for a proof)

1. $overline{fh} cong overline{fi}$ - given
2. $angle j cong angle g$ - given
3. $angle gfh cong angle ifj$ - given
4. $mangle hfj = mangle hfi + mangle ifj$ - reason to be filled
5. $mangle gfi = mangle gfh + mangle hfi$ - additive property of angle measure
6. $mangle hfj = mangle hfi + mangle gfh$ - substitution
7. $mangle gfi = mangle hfj$ - transitive property of equality
8. $\triangle fgi cong \triangle fjh$ - aas

Explanation:

Step1: Identify angle addition rule

The statement $m\angle HFJ = m\angle HFI + m\angle IFJ$ follows the Additive Property of Angle Measure, which states that the measure of a larger angle is the sum of the measures of its non-overlapping component angles.

Step2: Confirm congruence via AAS

We have $\angle J \cong \angle G$, $\angle GFI \cong \angle HFJ$ (from equal measures), and $\overline{FH} \cong \overline{FI}$. This matches the AAS (Angle-Angle-Side) congruence criterion, proving $\triangle FGI \cong \triangle FJH$.

Answer:

1. $\overline{FH} \cong \overline{FI}$ | Given
2. $\angle J \cong \angle G$ | Given
3. $\angle GFH \cong \angle IFJ$ | Given
4. $m\angle HFJ = m\angle HFI + m\angle IFJ$ | Additive Property of Angle Measure
5. $m\angle GFI = m\angle GFH + m\angle HFI$ | Additive Property of Angle Measure
6. $m\angle HFJ = m\angle HFI + m\angle GFH$ | Substitution
7. $m\angle GFI = m\angle HFJ$ | Transitive Property of Equality
8. $\triangle FGI \cong \triangle FJH$ | AAS

The missing reason for Step 4 is Additive Property of Angle Measure.