Question

complete the proof that $\triangle efh \cong \triangle eig$.

(image of triangle with points f, g, h, i, e and a table with statements and reasons)

1. $\overline{eh} \cong \overline{eg}$ - given
2. $\angle f \cong \angle i$ - given
3. $\angle feg \cong \angle hei$ - given
4. $m\angle gei = m\angle geh + m\angle hei$ - additive property of angle measure
5. $m\angle feh = m\angle feg + m\angle geh$ - additive property of angle measure
6. $m\angle gei = m\angle geh + m\angle feg$ -
7. $m\angle feh = m\angle gei$ -
8. $\triangle efh \cong \triangle eig$ -

(reasons for 6,7,8 include options like additive property of angle measure, all right angles are congruent, etc.)

Explanation:

Step1: Substitute congruent angles

From step 3, $\angle FEG \cong \angle HEI$, so $m\angle HEI = m\angle FEG$. Substitute into step 4:
$m\angle GEI = m\angle GEH + m\angle FEG$
Reason: Substitution Property of Equality

Step2: Set angle measures equal

From step 5 ($m\angle FEH = m\angle FEG + m\angle GEH$) and step 6, use transitive property:
$m\angle FEH = m\angle GEI$, so $\angle FEH \cong \angle GEI$
Reason: Transitive Property of Equality

Step3: Apply AAS congruence

We have $\angle F \cong \angle I$, $\angle FEH \cong \angle GEI$, $\overline{EH} \cong \overline{EG}$. Use Angle-Angle-Side congruence:
$\Delta EFH \cong \Delta EIG$
Reason: Angle-Angle-Side (AAS) Congruence Postulate

Answer:

Statement	Reason
2. $\angle F \cong \angle I$	Given
3. $\angle FEG \cong \angle HEI$	Given
4. $m\angle GEI = m\angle GEH + m\angle HEI$	Additive Property of Angle Measure
5. $m\angle FEH = m\angle FEG + m\angle GEH$	Additive Property of Angle Measure
6. $m\angle GEI = m\angle GEH + m\angle FEG$	Substitution Property of Equality
7. $m\angle FEH = m\angle GEI$	Transitive Property of Equality
8. $\Delta EFH \cong \Delta EIG$	Angle-Angle-Side (AAS) Congruence Postulate